Geometric function theory: the art of pullback factorization

Proposition 3.2. The collection of all p-integrable p-weak upper gradients of a map $u: X \to Z$ is a closed convex lattice inside $L^p(X)$ and, if nonempty, contains a unique element of smallest $L^p$ -norm. In particular, if a map has a p-integrable p-weak upper gradient, then it has a minimal p-weak upper gradient. In view of the above result, the minimal p-weak upper gradient $g_u$ should be thought of as a substitute for $|\nabla u|$ , or the length of a gradient, for functions defined in metric measure spaces. Fix a Banach space $\mathbb{V}$ , and we first define the Sobolev space $N^{1,p}(X,\mathbb{V})$ of $\mathbb{V}$ -valued mappings. Let $\tilde{N}^{1,p}(X,\mathbb{V})$ denote the collection of all maps $f \in L^p(X,\mathbb{V})$ that have an upper gradient in $L^p(X)$ . We equip it with seminorm $$||f||_{\tilde{N}^{1,p}(X,\mathbb{V})} = ||f||_{L^p(X,\mathbb{V})} + ||g_f||_{L^p(X)},$$ where $g_f$ is the minimal p-weak upper gradient of f. We obtain a normed space $N^{1,p}(X, \mathbb{V})$ by passing to equivalence classes of functions in $\tilde{N}^{1,p}(X, \mathbb{V})$ with respect to equivalence relation: $f_1 \sim f_2$ if $||f_1 - f_2||_{\tilde{N}^{1,p}(X,\mathbb{V})} = 0$ . Thus $$(3.4) N^{1,p}(X, \mathbb{V}) := \tilde{N}^{1,p}(X, \mathbb{V}) / \{ f \in \tilde{N}^{1,p}(X, \mathbb{V}) : ||f||_{\tilde{N}^{1,p}(X, \mathbb{V})} = 0 \}.$$ Let $\tilde{N}_{loc}^{1,p}(X,\mathbb{V})$ be the vector space of functions $f\colon X\to\mathbb{V}$ with the property that every point $x\in X$ has a neighborhood $U_x$ in X such that $f\in \tilde{N}^{1,p}(U_x,\mathbb{V})$ . Two functions $f_1$ and $f_2$ in $\tilde{N}_{loc}^{1,p}(X,\mathbb{V})$ are said to be equivalent if every point $x\in X$ has a neighborhood $U_x$ in X such that the restrictions $f_1|_{U_x}$ and $f_2|_{U_x}$ determine the same element in $\tilde{N}^{1,p}(U_x,\mathbb{V})$ . The local Sobolev space $N_{loc}^{1,p}(X,\mathbb{V})$ is the vector space of equivalent classes of functions in $\tilde{N}_{loc}^{1,p}(X,\mathbb{V})$ under the preceding equivalence relation. To define the Sobolev space $N^{1,p}(X,Y)$ of mappings $f: X \to Y$ , we first fix an isometric embedding $\varphi$ of Y into some Banach space $\mathbb{V}$ . Then the Sobolev space $N^{1,p}(X,Y)$ consists of all mappings $f: X \to Y$ with $\varphi \circ f \in N^{1,p}(X,\mathbb{V})$ . 3.3. Poincaré inequalities and spaces of locally bounded geometry. The following concept of an abstract Poincaré inequality was first introduced by Heinonen and koskela [40] and it plays an important role in the study of analysis on metric spaces; see for instance [32, 42].

Lemma 3.4. A complete and doubling metric measure space that supports a Poincaré inequality is quasiconvex, quantitatively. Recall that a metric space $Z = (Z, d_Z)$ is said to be C-quasiconvex or simply quasiconvex, $C \geq 1$ , if each pair of points can be joined by a rectifiable curve in Z such that $$(3.6) l(\gamma) \le C d_Z(x, y).$$ Recall that a metric space X is said to be $\theta$ -linearly locally connected ( $\theta$ -LLC) if there exists $\theta \ge 1$ such that for $x \in X$ and $0 < r \le \text{diam}(X)$ , $(\theta$ -LLC-1) every two points $a, b \in B(x, r)$ can be joined in $B(x, \theta r)$ , and (θ-LLC-2) every two points a, b ∈ X\B(x, r) can be joined in X\B(x, θ⁻¹ r). Here, by joining a and b in B we mean that there exists a continuum γ : [0, 1] → B with γ(0) = a, γ(1) = b. The following result was proved in [\[40,](#page-89-3) Corollary 5.8], where the quasiconvexity condition is provided by Lemma [3.4.](#page-15-1)

Proposition 3.5. Let (X, d, µ) be a complete Ahlfors Q-regular metric measure space that supports a (1, Q)-Poincar´e inequality. Then X is θ-linearly locally connected with θ depending only on the data associated with X. We next introduce an important class of metric spaces, where a large part of the theory of quasiconformal or/and quasiregular mappings can be extended as in the Euclidean spaces. The importance of such spaces was first realized in [\[40\]](#page-89-3) in their characterization of Poincar´e inequalities.

Lemma 4.1. If S ⊂ X and ψ −1 (ψ(S)) = S, then the restriction ψ|^S : S → ψ(S) is discrete and open as well with respect to the subspace topologies of S and ψ(S). Fix a normal subset D ⊂ X. For 1 ≤ n ≤ N(ψ, D), let Dⁿ = {x ∈ D : N(ψ(x), ψ, D) = n}. For each z ∈ ψ(D), we define M^z by $$M_z = \min \Big\{ \inf_{x, x' \in \psi^{-1}(z) \cap D} \frac{d(x, x')}{6}, \ d(\psi^{-1}(z), \ X \backslash D) \Big\}.$$ For each r > 0, we define U(x, ψ, r) ² = ψ −1 (B(ψ(x), r)) ∩ B(x, Mψ(x)). The following proposition allows us to choose radii R(z) = R(z, ψ, D) such that the sets U(x, ψ, r), x ∈ ψ −1 ({z}), behave reasonably well, provided r < R(ψ(x)). It is well-known to experts, but for completeness, we include the proofs here.

Proposition 4.2. Let D ⊂ X be normal. Then there is a function R: ψ(D) → (0, ∞) with the following properties: - (1) For each z ∈ ψ(D) and r ≤ R(z), B(z, R(z)) ⊂ ψ(D); - (2) For ever z ∈ ψ(D) and r ≤ R(z), $$\psi^{-1}\big(B(z,r)\big)\cap D=\bigcup_{x\in\psi^{-1}(z)\cap D}U(x,\psi,r);$$ (3) For every z ∈ ψ(D), r ≤ R(z), and z ′ ∈ B(z, r) $$N(z',\psi,D) = \sum_{x \in \psi^{-1}(z) \cap D} N(z',\psi,U(x,\psi,r)).$$ In particular, N(z, ψ, U(x, ψ, r)) = 1 for each x ∈ ψ −1 (z) ∩ D; - (4) For every z ∈ ψ(D), r ≤ R(z), and each x ∈ ψ −1 (z), ψ(U(x, ψ, r)) = B(z, r); - (5) For every z ∈ ψ(D), r ≤ R(z), and each z ′ ∈ B(z, r), N(z ′ , ψ, D) ≥ N(z, ψ, D); - (6) For each z ∈ ψ(D), r ≤ R(z), and each x ∈ ψ −1 (z), ψ is injective on Dn∩U(x, ψ, r); - (7) For each x, x′ ∈ Dn, each r ≤ R(ψ(x)), and each r ′ ≤ R(ψ(x ′ )), if B(ψ(x), r) ⊂ B(ψ(x ′ ), r′ ), then either ψ is injective on $$D_n \cap (U(x, \psi, r) \cup U(x', \psi, r'))$$ or $U(x, \psi, r) \cap U(x', \psi, r') = \emptyset;$ ²We will also use U(x, ψ, r) to denote the x-component of ψ −1 (B(ψ(x), r)) in later sections. (8) For each $x, x' \in D$ , $r \leq R(\psi(x))$ , and each $r' \leq R(\psi(x'))$ , if $B(\psi(x), r) \subset B(\psi(x'), r')$ , then either $$U(x, \psi, r) \subset U(x', \psi, r')$$ or $U(x, \psi, r) \cap U(x', \psi, r') = \emptyset$ . Moreover, every other function $\tilde{R}$ : $\psi(D) \to (0, \infty)$ such that $\tilde{R}(z) \leq R(z)$ for each $z \in \psi(D)$ satisfies all these properties as well.

Lemma 4.3. Let $1 \le n \le N(\psi, D)$ . Then there are pairwise disjoint Borel subsets $D_{n,j}$ , $j = 1, \ldots, n$ , such that $D_n = \bigcup_{j=1}^n D_{n,j}$ and such that for each j, $\psi|_{D_{n,j}}$ is a bijection onto $\psi(D_n)$ . Proof. Let $\{V_i\}$ be a countable cover of $D_n$ by open subsets $V_i \subset D$ , such that $\psi|_{D_n \cap V_i}$ is injective. Such a cover exists by the relative compactness, and hence separability of D, along with Proposition 4.2 (6). For each $j = 0, \ldots, n$ , construct $D_{n,j}$ as follows: Let $D_{n,0} = \emptyset$ . Having defined $D_{n,j}$ for all $j < k \le n$ , let $D_{n,k}^0 = \emptyset$ , and for each integer $i \ge 1$ , define $D_{n,j}^i$ by $$D_{n,j}^{i} = D_{n,j}^{i-1} \cup \left( V_{i} \setminus \left( \psi^{-1} \left( \psi(D_{n,j}^{i-1}) \right) \cup \bigcup_{j=0}^{k-1} D_{n,j} \right) \right).$$ Finally, let $D_{n,k} = \bigcup_{i=1}^{\infty} D_{n,j}^{i}$ . By construction, the sets $D_{n,j}$ are disjoint Borel sets on which $\psi$ is injective. It remains to show that $\psi(D_{n,j}) = \psi(D_n)$ . To see this, fix $1 \le k \le n$ and $z \in \psi(D_n)$ . The union $\bigcup_{j=1}^{k-1} D_{n,j}$ contains at most (k-1) elements of the fiber $\psi^{-1}(z)$ , so there is some $x \in D_n \setminus \bigcup_{j=1}^{k-1} D_{n,j}$ such that $\psi(x) = z$ . Since the sets $V_i$ are an open cover of $D_n$ , there is some $V_i \ni x$ . Then either $z \in \psi(D_{n,k}^{i-1})$ or $z \notin \psi(D_{n,k}^{i-1})$ . In the latter case, $x \notin \psi^{-1}(\psi(D_{n,k}^{i-1}))$ . By definition, we have $x \in D_{n,k}^{i}$ , and so $z = \psi(x) \in \psi(D_{n,k})$ . $\square$ 4.2. The pullback measure. Let $\rho: X \to [0, \infty)$ be a function. We define two functions $\sup(\rho, y, \psi, A)$ and $\sum(\rho, y, \psi, A)$ with respect to a branched covering $\psi: X \to Y$ by $$\sup(\rho,y,\psi,A) = \sup_{x \in \psi^{-1}(y) \cap A} \rho(x)$$ and $$\sum (\rho, y, \psi, A) = \sum_{x \in \psi^{-1}(y) \cap A} \rho(x).$$ Suppose, as we will do from now on, that Y is equipped with a (locally finite, Borel-regular) measure $\nu$ . We define the "pullback measure" $\psi^*\nu$ on X via the formula $$\psi^*\nu(A) = \int_Y N(y, f, A) d\nu(y).$$ By the subadditivity of the integral, $\psi^\nu$ is an outer measure. Note that by definition, for every subset $A \subset X$ , $\psi^\nu(A) = 0$ if and only if $\nu(\psi(A)) = 0$ . Since the restrictions $\psi|_{D_n}$ are local homeomorphisms, it follows that $\psi$ maps Borel sets to Borel sets, and that $\psi^\nu$ is a locally finite Borel regular outer measure on X. Let $A \subset X$ . Since $\psi$ preserves Borel sets, and $\nu(\psi(A)) = 0$ if and only if $\psi^\nu(A) = 0$ , it follows from the Borel regularity of each measure that A is $\psi^\nu$ -measurable if and only if $\psi(A)$ is $\nu$ -measurable. Thus, if $\rho \colon X \to \mathbb{R}$ is a Borel (resp. $\psi^\nu$ -measurable) function on X and $A \subset X$ a Borel set, then $\sup(\rho, \cdot, \psi, A)$ and $\sum(\rho, \cdot, \psi, A)$ are Borel (resp. $\nu$ -measurable) functions on Y. Moreover, by standard approximation arguments, we have (4.1) $$\int_{X} \rho(x)d\psi^*\nu(x) = \int_{Y} \sum_{x} (\rho, y, \psi, X)d\nu(y),$$ for every Borel function $\rho: X \to \mathbb{R}$ . We define the (volume) Jacobian of $\psi \colon X \to Y$ by $$J_{\psi} := \frac{d\psi^* \nu}{d\mu},$$ and the inverse (volume) Jacobian of $\psi$ by $$J_{\psi}^{-1} := \frac{d\mu}{d\psi^*\nu}.$$ We say that $\psi$ satisfies Condition N if $\nu(\psi(A)) = 0$ whenever $\mu(A) = 0$ , and Condition $N^{-1}$ if $\mu(A) = 0$ whenever $\nu(\psi(A)) = 0$ . Since, by the preceding discussion, $\psi^\nu(A) = 0$ if and only if $\nu(\psi(A)) = 0$ , Condition N is equivalent to the condition that $\psi^\nu \ll \mu$ , and Condition $N^{-1}$ is equivalent to the condition that $\nu \ll \psi^*\nu$ . From this, it immediately follows that (4.2) $$\int_X \rho(x) J_{\psi}(x) d\mu(x) \le \int_X \rho(x) d\psi^* \nu(x) = \int_Y \sum (\rho, y, \psi, X) d\nu(y),$$ for every Borel function $\rho: X \to \mathbb{R}$ , and the two sides are equal for every Borel function $\rho$ if and only if $\psi$ satisfies Condition N. Similarly, (4.3) $$\int_{Y} \sum (\rho J_{\psi}^{-1}, y, \psi, X) d\nu(y) = \int_{X} \rho(x) J_{\psi}^{-1}(x) d\psi^{*} \nu(x) \le \int_{X} \rho(x) d\nu(x),$$ with equality for all $\rho$ if and only if $\psi$ satisfies Condition $N^{-1}$ . 4.3. Integration theory in pullback geometry. As already mentioned in the introduction, we will later use certain pullback normal neighborhoods, instead of balls, to run the covering arguments from [4] to relate metrically quasiregularity and analytic quasiregularity. From the technical point of view, it is important that our special family of normal neighborhoods will have many nice behaviors as the family of balls. For this reason, we develop a variant of the Lebesgue-Radon-Nikodym theory in this section.

Lemma 4.4. Let U be a family of sets of the form U = U(x, ψ, r), with x ∈ D and 5r^x < R(ψ(x)), and suppose ν is locally doubling. Then there is a countable, pairwise disjoint subfamily U ′ ⊂ U such that $$\bigcup_{U \in \mathcal{U}} U \subset \bigcup_{U' \in \mathcal{U}'} 5U'.$$ Proof. If U is empty the theorem is trivial. Otherwise, we construct U ′ = {Ui}, where each Uⁱ = U(xⁱ , ψ, ri) ∈ U, in the following manner. For ease of notation, we index the family U by some index set A, so that U = S ^α∈A Uα, where U^α = U(xα, ψ, rα). We construct U ′ inductively. We first choose U⁰ ∈ U such that $$r_0 \ge \frac{1}{2} \sup_{\alpha \in \mathcal{A}} r_{\alpha}.$$ Having chosen U^k for all k < n, let $$\mathcal{A}_n = \left\{ \alpha \in \mathcal{A} : U_\alpha \nsubseteq \bigcup_{k=1}^{n-1} 5U_k \right\}$$ . If Aⁿ is nonempty, choose Uⁿ such that $$r_n \ge \frac{1}{2} \sup_{\alpha \in \mathcal{A}_n} r_{\alpha}.$$ We proceed inductively, choosing Uⁿ for each n ∈ N, unless Aⁿ is empty for some n, in which case the sequence terminates at n − 1. For each i, let Bⁱ = B(ψ(xi), ri) = ψ(Ui). We first note that by construction, for m < n, Uⁿ \ 5Um, and rⁿ ≤ 2rm, so that diam(Bn) ≤ 4rm. Now, Bⁿ \ 5Bm, and so Bⁿ ∩ B^m = ∅, from which it follows that Uⁿ ∩ U^m = ∅. It remains to show the inclusion. If the sequence terminates, this is immediate. Otherwise, we need only show that T^∞ ⁿ=1 Aⁿ = ∅. Suppose by contradiction that α ∈ T^∞ ⁿ=1 An. By our construction, 0 < r^α ≤ 2rⁿ for each n ∈ N. By the doubling property of ν and the compactness of D, there is a constant C > 0 such that ψ ^∗ν(Un) ≥ ν(Bn) ≥ C for each n. Since the sets Uⁿ are disjoint, this implies that $$\psi^ \nu(D) \ge \sum_{n=1}^{\infty} \psi^ \nu(U_n) = \infty,$$ contradicting the local finiteness of ψ ^∗ν. Remark 4.5. The assumption that ν is locally doubling is not necessary; we have assumed it since it is harmless and makes the proof somewhat more straightforward. In fact, if D has finite topological dimension N, then by the Whitney embedding theorem, D is homeomorphic to a subset D′ ⊂ R ²N+1. Thus we may assume that D is a subset in R ²N+1. By [\[34\]](#page-89-13), D may be equipped with a doubling measure. Moreover, even in the case when the topological dimension of D is infinite, the theorem can be proved with an argument along the lines of [\[34,](#page-89-13) Proof of Theorem 1.2], with the countability of U ′ following from the compactness of D. Inspired by the covering lemmas from [\[4\]](#page-88-3), we introduce the following concept.

Lemma 4.7. Let $\mathcal{A} = \{A_i : x_i \in A_i\}_{i \in I}$ be an admissible sequence of pointed sets such that for each $k \in I$ , $$B(x_k, r_k) \subset A_k \subset B(x_k, Hr_k).$$ Then for each $i \neq j$ in I, $B(x_i, r_i/5H) \cap B(x_j, r_j/5H) = \emptyset$ .

Lemma 4.8. Let X be a metric space, $S \subset X$ relatively compact, and for each $x \in S$ , let $A_x \subset X$ and $r_x > 0$ satisfy $$(4.4) B(x, r_x) \subset A_x \subset B(x, Hr_x).$$ Then there is a sequence of points $x_i \in S$ such that the corresponding sequence $A_i = A_{x_i}$ is admissible, and $$(4.5) S \subset \bigcup_{i=1}^{\infty} A_i.$$

Lemma 4.9. Let $D \subset X$ be a normal domain, $A \subset D$ a Borel set, and $\mathcal{U}$ a family of sets of the form $U = U(x, \psi, r)$ , such that for each $x \in A$ , there is a sequence $r_i$ converging to 0 such that $U(x, \psi, r_i) \in \mathcal{U}$ . Then for each $\varepsilon > 0$ there is a pairwise disjoint, countable family of sets $U_i = U(x_i, \psi, r_i)$ such that $$\psi^*\nu\Big(A\backslash\bigcup_{i=1}^\infty U_i\Big)<\varepsilon.$$

Corollary 4.10. If $\rho \in L^p_{loc}(\psi^\nu)$ with a doubling measure $\nu$ . Then $\psi^\nu$ -a.e. $x \in X$ is a $(p, \psi)$ -Lebesgue point of $\rho$ .

Corollary 4.11. Let $\mu$ be a Radon measure on X, and let $\nu$ be doubling. Then the Radon-Nikodym derivative of $\mu$ with respect to $\psi^\nu$ is given, at $\psi^\nu$ -a.e. $x \in X$ , by $$\frac{d\mu}{d\psi^\nu}(x) = \lim_{r \to 0} \frac{\mu(U(x, \psi, r))}{\psi^\nu(U(x, \psi, r))} = \lim_{r \to 0} \frac{\mu(U(x, \psi, r))}{\nu(B(\psi(x), r))}.$$

Lemma 4.12. It always holds that 1 ≤ iess(x, ψ) ≤ i(x, ψ). Moreover, if x is a ψ-Lebesgue point for Dn, where Dⁿ = {z ∈ D : N(ψ(z), f, D) = n}, then iess(x, ψ) = 1. In particular, if ν is doubling, then iess(x, ψ) = 1 ψ ^∗ν-a.e. in X. Proof. The inequalities 1 ≤ iess(x, ψ) ≤ i(x, ψ) follows trivially from the definition and the fact that 1 ≤ N(y, ψ, U(x, ψ, r)) ≤ i(x, ψ) for all r sufficiently small and all y ∈ B(ψ(x), r). If x is a ψ-Lebesgue point for Dn, then $$\lim_{r\to 0} \frac{\psi^\nu(U(x,\psi,r))}{\psi^\nu(U(x,\psi,r)\cap D_n)} = 1,$$ which implies, by definition and Proposition [4.2](#page-18-2) (6), that iess(x, ψ) = 1. The last assertion follows immediately from Corollary [4.10.](#page-25-1)

Lemma 5.4. Let $\pi: X \to Y$ be a proper branched covering and let $\gamma: [a, b] \to Y$ be a path. Then for each $x \in \pi^{-1}(\gamma(a))$ , there is a curve $\tilde{\gamma}: [a, b] \to X$ such that $\tilde{\gamma}(a) = x$ and $\gamma = \pi \circ \tilde{\gamma}$ . Another important fact (see e.g. Proposition 4.2) which we will use repeatedly is that for each $x \in X$ , the sets $U(x, \pi, r)$ , r > 0, form a neighborhood basis of z in the topology of X, where from now on, $U(x, \pi, r)$ denotes the x-component of $\pi^{-1}(B(\pi(x), r))$ . Remark 5.5. i). For simplicity, we are not working in quite as much generality as we could. In fact, it would often suffice for our purposes to work in the generality of light open mappings. However, locally finite (and indeed, locally bounded) multiplicity is necessary for a number of our strongest results. Moreover, the theory of BLD mappings (and more generally, quasiregular mappings) between manifolds, or even generalized manifolds, typically requires $\pi$ to be light, open and sense-preserving (or sense-reversing), which in that setting is equivalent to the condition that $\pi$ is discrete, open, and of locally bounded multiplicity [45]. ii). It should be noted, however, that in general, local compactness of Z and discreteness of $\pi$ imply locally finite multiplicity (i.e., each $x \in X$ has a neighborhood U such that for all $y \in Y$ , $N(y, \pi, U) < \infty$ ), but not locally bounded multiplicity. For example, let $$X = Y = \bigcup_{i=1}^{\infty} \left\{ (t, it) : 0 \le t \le \frac{1}{i^2} \right\} \subset \mathbb{R}^2,$$ with X equipped with the subspace topology and Y inheriting the Euclidean metric from $\mathbb{R}^2$ . Let $\pi\colon X\to Y$ be given by $\pi(t,it)=(\frac{i^2t}{k^2},\frac{i^2t}{k})$ whenever $2^k\le i<2^{k+1}$ . Then X is compact, $\pi$ is discrete and open, and for every neighborhood $U\subset X$ of the origin, we have $N(\pi,U)=\infty$ , even though for each $y\in Y$ , $N(y,\pi,U)<\infty$ . Remark 5.6. It is an elementary topological fact that a locally compact, connected, locally connected metric space is path connected, and thus every connected open subset is path connected as well. It also follows from a simple diagonalization argument that locally, the infimum in inequality (5.1) is in fact attained, i.e., at every $x \in X$ , there is a neighborhood U of x such that for every $x' \in U$ , there is a continuum $\alpha$ joining x and x' with $$f^*d_Y(x, x') = \operatorname{diam}(f(\alpha)).$$ It is also perhaps of interest that by the continuity and openness of f, every continuum $\alpha$ has a connected open (and a fortiori path connected) neighborhood $\alpha_{\varepsilon}$ with $$\operatorname{diam}(f(\alpha_{\varepsilon})) \leq \operatorname{diam}(f(\alpha)) + \varepsilon$$ for each $\varepsilon > 0$ . Thus, in the definition of pullback metric, we could just as well have required $\alpha$ to be (the image of) a curve, or a connected open set, though in the latter case the infimum in (5.1) need not necessarily be realized. For simplicity, we will formulate many of our basic results for the case that $\pi$ is proper (i.e. the preimage of each compact set is compact) and surjective, and $N(\pi, X) < \infty$ . We lose very little generality with this reduction, in light of the following considerations. First, if $\pi$ is an arbitrary branched covering, then at every $x \in X$ , there is a radius r > 0 for which $U(x, \pi, r)$ is relatively compact. From that and the local boundedness of the multiplicity of $\pi$ , we have $$N(\pi|_{U(x,\pi,r)}, U(x,\pi,r)) = N(\pi, U(x,\pi,r)) < \infty.$$ Since $\pi$ is open, $\pi(\partial U(x,\pi,r)) = \partial \psi(U(x,\pi,r))$ , so that $$\pi|_{U(x,\pi,r)} \colon U(x,\pi,r) \to \pi(U(x,\pi,r))$$ is proper and surjective as well, and so local restrictions of $\pi$ satisfy the more restrictive conditions. Secondly, it follows easily from the local connectivity of X that for any two open subsets $U, V \subset X$ , the pullback metrics $(\pi|_U)^d_Y$ and $(\pi|_V)^d_Y$ are locally isometric on $U \cap V$ (see e.g. Lemma 5.10 below). As a result of the above considerations, when analyzing the metric space $X^{\pi}$ , it is often enough to apply our following results to the metric spaces $(\pi|_{U(x,\pi,r)})^*Y$ , and then invoke the local isometries between these spaces and $X^{\pi}$ . Up to the end of this section, we will assume the metric space Y is proper, i.e. closed bounded balls are compact.

Lemma 5.7. The metric space $X^{\pi}$ is a proper metric space, homeomorphic to X via the indentity mapping g. Open and closed balls in $X^{\pi}$ are connected and $X^{\pi}$ have 1-bounded turning. The projection mapping $\pi \colon X^{\pi} \to Y$ is 1-Lipschitz, 1-BDD and for each $z \in X^{\pi}$ , $$(5.3) B(z,r) \subset U(z,\pi,r) \subset B(z,2r).$$

Lemma 5.8. If Y has c-bounded turning, then $\pi$ is c-co-Lipschitz and is locally c-bi-Lipschitz on each set $X_k := \{z \in X^{\pi} : N(z, \pi, X) = k\}$ . If, additionally, Y is Ahlfors Q-regular with constant $c_2$ , then $X^{\pi}$ is Ahlfors Q-regular with constant $c^Q c_2 N$ . Moreover, for each k = 1, ..., N, and at each Lebesgue point z of $X_k$ , we may take the pointwise constant of Q-regularity to be $c^Q c_2$ . Proof. If Y has c-bounded turning, then for each $z_0 \in X^{\pi}$ and r > 0, every point $y \in B(\pi(z_0), r)$ may be joined to $\pi(z_0)$ with a continuum $\alpha$ of diameter at most cr. We have by Lemma 5.4 that there is a continuum $\tilde{\alpha}$ containing $z_0$ such that $\pi(\tilde{\alpha}) = \alpha \ni y$ . Thus there is some point $z \in \alpha$ with $\pi(z) = y$ , so that $$\pi^* d_Y(z_0, z) \le \operatorname{diam}(\tilde{\alpha}) = \operatorname{diam}(\alpha) \le cr.$$ Thus $y \in \pi(B(z_0, cr))$ and so $\pi$ is c-co-Lipschitz. It follows easily from local compactness and Proposition 4.2 (5) that the multiplicity function $N(y, \pi, X^{\pi})$ is lower-semicontinuous in Y, which in turn implies that $\pi$ is locally bijective on each of the sets $X_k$ . From this and the fact that $\pi$ is 1-Lipschitz and c-co-Lipschitz we obtain that $\pi$ is locally c-bi-Lipschitz on each $X_k$ . We next turn to the second claim. Note that $\pi$ is 1-Lipschitz, c-co-Lipschitz and that Y is Ahlfors Q-regular with constant $c_2$ , so we have $$\mathcal{H}^Q(B(x,r)) \ge \mathcal{H}^Q(\pi(B(x,r))) \ge \mathcal{H}^Q(B(\pi(x),r/c)) \ge \frac{1}{c_2 c_2^Q} r^Q.$$ For the reverse direction, we first assume that $\pi$ is c-bi-Lipschitz on $X_k$ and write $B(x,r) = \bigcup_{k=1}^N B_k(x,r)$ , where $B_k(x,r) = B(x,r) \cap X_k$ . Then it follows from the subadditivity of the Hausdorff measure that $$\mathcal{H}^{Q}(B(x,r)) \leq \sum_{k=1}^{N} \mathcal{H}^{Q}(B_{k}(x,r)) \leq N \max_{1 \leq k \leq N} \mathcal{H}^{Q}(B_{k}(x,r)).$$ For each k, note that $\pi^{-1}$ : $\pi(B_k(x,r)) \to B_k(x,r)$ is c-Lipschitz and that $\pi$ is 1-Lipschitz, and we conclude $$\mathcal{H}^Q(B_k(x,r)) = \mathcal{H}^Q(\pi^{-1} \circ \pi(B_k(x,r))) \le c^Q \mathcal{H}^Q(\pi(B(x,r))) \le c^Q \mathcal{H}^Q(B(\pi(x),r)).$$ In the general case, we may use Lemma 4.3 to decompose each $B_k(x,r)$ as a disjoint union of Borel sets on which $\pi$ is c-bi-Lipschitz. Then a similar computation leads to the estimate $\mathcal{H}^Q(B(x,r)) \leq Nc_2c^Qr^Q$ . In conclusion, we have shown $$\frac{1}{c_2 c^Q} r^Q \le \mathcal{H}^Q(B(x,r)) \le N c_2 c^Q r^Q.$$ At each Lebesgue point z of $X_k$ , by Proposition 4.2, $\pi$ is injective in a sufficiently small neighborhood of z and so we may repeat the above argument with N=1, which gives the desired pointwise constant $c^Q c_2$ . For the next result, recall that a point x in a metric space X is called a local cut point if $U\setminus\{x\}$ is disconnected for some neighborhood U of x.

Lemma 5.9. If Y is c-LLC-2 and X has no local cut points, then $X^{\pi}$ is locally 2c-LLC. Proof. By Lemma 5.7, we only need to check the LLC-2 property of $X^{\pi}$ . Since X has no local cut points, so is $X^{\pi}$ . It follows that for each $z_0 \in X^{\pi}$ and $r_0 > 0$ such that $\pi^{-1}(\pi(z_0)) \cap \overline{U(z_0, \pi, r_0)} = \{z_0\}$ , there is some $R < \frac{r_0}{2}$ such that for each $z \in U(z_0, \pi, \frac{r_0}{2})$ and r < R, every pair of points $z'_1, z'_2 \in \partial U(z_0, \pi, r_0)$ can be joined by a continuum $\alpha$ such that $\alpha$ is disjoint from $U(z, \pi, r)$ . Therefore, let $z \in U(z_0, \pi, \frac{r_0}{2})$ and r < R, and suppose $z_1, z_2 \in X^{\pi} \setminus U(z, \pi, r)$ . Let $$y_0 = \pi(z_0)$$ , $y = \pi(z)$ , and $y_i = \pi(z_i)$ for $i = 1, 2$ . First suppose additionally that $y_1, y_2 \in Y \setminus B(y, r)$ . Then by the LLC-2 property, we may join $y_i$ to $\partial B(y_0, r_0)$ with a continuum $\alpha_i \subset Y \setminus B(y, r/c)$ , for i = 1, 2. By Lemma 5.4, we may lift each continuum $\alpha_i$ to a continuum $\tilde{\alpha}_i$ containing $z_i$ , which clearly intersects $\partial U(z_0, \pi, r_0)$ , say at $z_i'$ . Since r < R, there is a continuum $\alpha' \subset X^{\pi} \setminus U(z, \pi, r)$ joining $z_1'$ and $z_2'$ , and so $$\tilde{\alpha}_1 \cup \alpha \cup \tilde{\alpha}_2 \subset X^{\pi} \backslash U(z, r/c)$$ is a continuum joining $z_1$ and $z_2$ . Since $B(z, r/c) \subset U(z, \pi, r/c)$ and $U(z, \pi, r) \subset B(z, 2r)$ , we conclude that $X^{\pi}$ is locally 2c-LLC. Next, suppose $y_1 \in B(y,r)$ and $y_2 \in Y \setminus B(y,r)$ . Then $U(z_1,\pi,r) \cap U(z,\pi,r) = \emptyset$ . Let $\alpha_2$ , $\alpha'_2$ and $z'_2$ be as in the preceding proof. It is clear that we only need to construct a continuum $\alpha_1$ containing $y_1$ so that its lift $\alpha'_1$ is disjoint from $U(z,\pi,r)$ and intersects $\partial U(z_0,\pi,r_0)$ at $\tilde{z}'_1$ . We first select a continuum $\beta_1$ that joins $y_1$ to some point $y'_1 \in \partial B(y,r)$ so that its lift $\beta'_1$ is disjoint from $U(z,\pi,r)$ and joins $z_1$ to some point $z'_1 \in \partial U(z_1,\pi,r)$ . Then since $y'_1 \in Y \setminus B(y,r)$ , we may join $y'_1$ to some point $\tilde{z}_1 \in \partial B(y_0,r_0)$ with a continuum $\gamma_1 \subset Y \setminus B(y,r/c)$ . As before, we may lift $\gamma_1$ to a continuum $\gamma'_1$ containing $z'_1$ , which clearly intersects $\partial U(z_0, \pi, r)$ , say at $\tilde{z}'_1$ . Then $\alpha'_1 := \beta'_1 \cup \gamma'_1$ will be a continuum joining $z_1$ to $\partial U(z_0, \pi, r_0)$ that is disjoint from $U(z, \pi, r/c)$ . Finally, suppose $y_1, y_2 \in B(y, r)$ . We may repeat the preceding arguments for both $y_1$ and $y_2$ to obtain two continua $\alpha'_1$ and $\alpha'_2$ that are disjoint from $U(z, \pi, r/c)$ . Moreover, for each $i = 1, 2, \alpha'_i$ joins $z_i$ to $\partial U(z_0, \pi, r_0)$ . We may repeat the arguments in the first case to complete the

Lemma 5.10. If $l_{d_Y}$ and $d_Y$ induces the same topology on Y, then the metrics $\pi^d_Y$ , $l_{\pi^d_Y}$ , $\pi^l_{d_Y}$ , and $l_{\pi^l_{d_Y}}$ induces the same topology on X. The length of a curve in $X^{\pi}$ is the same with respect to any of these four metrics, and moreover, $$\pi^ d_Y \le \pi^ l_{d_Y} \le l_{\pi^ d_Y} = l_{\pi^ l_{d_Y}} \le (2N - 1)\pi^* l_{d_Y}.$$ In particular, the metric space $\pi^*(Y^l)$ is (2N-1)-quasiconvex and if Y is c-quasiconvex, then $X^{\pi}$ is (2N-1)c-quasiconvex.

Proposition 5.12. f is L-BDD if and only if g is L-BDD. Proof. If g is L-BDD, then it follows immediately from the fact π is 1-BDD that f is L-BDD as well. For the reverse direction, simply notice that if g is not L-BDD, then there exists a non-constant curve γ ⊂ X such that either $$\operatorname{diam}(g(\gamma)) > L \operatorname{diam}(\gamma)$$ or $\operatorname{diam}(g(\gamma)) < \frac{1}{L} \operatorname{diam}(\gamma)$ . Taking into account the fact that π is 1-BDD, it means either $$\operatorname{diam}(\pi \circ g(\gamma)) > L \operatorname{diam}(\gamma)$$ or $\operatorname{diam}(\pi \circ g(\gamma)) < \frac{1}{L} \operatorname{diam}(\gamma)$ , which contradicts with our assumption f being L-BDD. Since X^f has 1-bounded turning, the latter implies that g −1 : X^f → X is L-Lipschitz. Indeed, for any z1, z² ∈ X^f , we may select a curve α ⊂ X^f , joining z¹ and z2, so that f ^∗d^Y (z1, z2) = diam(α). Then ˜α = g −1 (α) will be a curve in X that connects g −1 (z1) and g −1 (z2). Thus the L-BDD property of f and 1-BDD property of π give us $$d_X(g^{-1}(z_1), g^{-1}(z_2)) \le \operatorname{diam}(\tilde{\alpha}) \le L \operatorname{diam}(f(\tilde{\alpha})) = L \operatorname{diam}(\pi(\alpha))$$ $$= L \operatorname{diam}(\alpha) = L f^* d_Y(z_1, z_2).$$ Thus when g is L-Lipschitz and L-BDD, g is an L-bi-Lipschitz equivalence between X and X^f , whereby X has L-bounded turning. Conversely, if f is L-BDD, and X has c-bounded turning, then g is Lc-Lipschitz. Similar to the BDD case (with indeed the same proof, but using only the 1-BLD property of π), we have the following useful conclusion.

Proposition 5.13. f is L-BLD if and only if g is L-BLD. As in the BDD case, if g is L-BLD, then (g l ) −1 is L-Lipschitz, where g l : X → (X^f ) l is the identity, and so when f is L-Lipschitz and L-BLD, g ^l gives a bi-Lipschitz equivalence between X and (X^f ) l . Thus much of the theory of BLD and BDD mappings reduces to the study of the pullback metric. Similarly, it turns out that under various definitions and for many different levels of generality, g is quasiconformal if and only if f is quasiregular. Thus one obtains a canonical factorization of a quasiregular mapping into a composition of a quasiconformal mapping with a 1-BDD mapping. This is particularly useful in extending the theory of quasiregular mappings to the metric setting, as the quasiconformal theory has at present advanced much further than its branched counterpart in this generality. In fact, one of the motivations for our exploration of the pullback metric is to establish the equivalence of various geometric, metric, and analytic characterizations of quasiregularity, which we explore in Section [6;](#page-39-0) this equivalence has been established already in various contexts for definitions involving "outer" dilatation, (e.g., the geometric KO-inequality) which tend to mimic the proofs in the quasiconformal case, but the problem is substantially more delicate, even in the classical case, when inner dilatation is concerned.

Proposition 5.14. Suppose that Y is c-quasiconvex, and that f is L-Lipschitz and L-BLD with N = N(f, X) < ∞. Then f is cNL-BDD. Proof. Since f is L-Lipschitz, diam(f(α)) ≤ L diam(α) for each continuum α ⊂ X. On the other hand, by the above discussion, g yields an L-bi-Lipschitz equivalence between the length space X^l and (π ^∗Y ) , and so by Lemma [5.10,](#page-36-0) we have $$\operatorname{diam}(\alpha) \leq \operatorname{diam}_{l_{d_X}}(\alpha) \leq L \operatorname{diam}_{l_{\pi^*Y}}(g(\alpha)) \leq cNL \operatorname{diam}(g(\alpha)) = cNL \operatorname{diam}(f(\alpha)),$$ whereby $f$ is $cNL$ -BDD. $\Box$

Theorem 6.9. Let $f: \tilde{\Omega} \to \Omega$ be a branched covering. Then f is analytically $K_O$ -quasiregular with exponent Q if and only if it is geometrically $K_O$ -quasiregular with exponent Q. Similarly, f is inverse analytically $K_I$ -quasiregular with exponent Q if and only if it is strong inverse geometrically $K_I$ -quasiregular with exponent Q. The latter assertion in Theorem 6.9 provides us a useful analytic characterization of the Poletsky's inequality, which will be crucial in our proof of "the metric definition implies all the others". This characterization also gives us an alternative way of showing the Poletsky's inequality for quasiregular mappings, namely, we just need to verify that the lifting mapping q from the pullback factorization satisfies the inverse analytic definition of quasiconformality. The advantage of this alternative approach is that the metric/geometric information of the underlying metric measure spaces in practice is inherited very well at the level of its lifting space and is often very elementary to verify. There is no surprise that it is much easier to deal with homeomorphisms than the more general branched coverings. However, we caution the readers that the information on the branch set will be transferred to the pullback measure of the lifting space. Thus, instead of analyzing (uniform) Ahlfors regular spaces, we have to deal with pointwise Ahlfors regular spaces; see Section 6.4 below for a precise meaning. 6.2.1. Auxiliary results. We will need the following result later, in order to prove that f is geometrically $K_I$ -quasiregular with exponent Q if and only if g is geometrically $K_I$ -quasiconformal with exponent Q.

Lemma 6.10. Let $\rho: \tilde{\Omega}^f \to \mathbb{R}$ be a Borel function and let $\xi$ be a locally finite Borel regular measure on $\tilde{\Omega}^f$ . Suppose that for every open subset $\tilde{\Omega}_1^f \subset \tilde{\Omega}^f$ (not necessarily connected) such that $N(\pi, \tilde{\Omega}_1^f) < \infty$ , $$\int_{\Omega} \sup(\rho, y, \pi, \tilde{\Omega}^f) d\nu(y) \le \xi(\tilde{\Omega}_1^f).$$ Then $\rho \leq \frac{d\xi}{d\lambda} \lambda$ -a.e. in $\tilde{\Omega}^f$ .

Lemma 6.11. Let $\tilde{\Omega}_1^f \subset \tilde{\Omega}^f$ and let $\Gamma$ be a family of curves in $\tilde{\Omega}_1^f$ . Then for every Borel function $\rho$ admissible for $\pi(\Gamma)$ , $$\operatorname{Mod}_Q(\Gamma) \leq \int_{\Omega} \rho^Q(y) N(y, \pi, \tilde{\Omega}_1^f) d\nu(y).$$

Lemma 6.12. Let $\Gamma$ and $\Gamma'$ be families of curves in $\tilde{\Omega}^f$ and $\Omega$ , respectively. Suppose for each $\gamma' \in \Gamma'$ , there are curves $\gamma_1, \ldots, \gamma_m \in \Gamma$ and subcurves $\gamma'_1, \ldots, \gamma'_m$ of $\gamma'$ such that for each $i = 1, \ldots, m, \ \gamma'_i = \pi(\gamma_i)$ , and for a.e. $s \in [0, l(\gamma')], \ \gamma_i(s) = \gamma_j(s)$ if and only if i = j. Then $$m \operatorname{Mod}_Q(\Gamma') \leq \operatorname{Mod}_Q(\Gamma).$$ In particular, for every $\Gamma \subset \tilde{\Omega}^f$ , $$\operatorname{Mod}_{\mathcal{Q}}(\pi(\Gamma)) \leq \operatorname{Mod}_{\mathcal{Q}}(\Gamma).$$

Corollary 6.13. A curve family $\Gamma$ of curves in $\tilde{\Omega}^f$ is exceptional if and only if $\pi(\Gamma)$ is.

Proposition 6.14. Let $\rho \in L^p_{loc}(\Delta)$ and $h: \Delta \to W$ be a continuous mapping. Then $\rho$ is a p-weak upper gradient of h if and only if for p-almost every curve $\gamma: [a, b] \to \Delta$ , the following two conditions are satisfied: - h is absolutely continuous along $\gamma$ ; - If the parametrization of $\gamma$ itself is absolutely continuous as well, then the inequality (6.4) $$\rho(\gamma(t))v_{\gamma}'(t) \ge v_{h(\gamma)}'(t)$$ holds a.e. on the parametrizing interval [a, b]. Note that in the previous proposition every rectifiable curve (and hence p-almost every curve) has an absolutely continuous parametrization, namely, the arc-length parametrization. Let $C_{\varepsilon}(W)$ be the collection of curves $\gamma \colon [a,b] \to W$ such that $d(\gamma(a),\gamma(b)) \geq \varepsilon$ . For every mapping $h \colon \Delta \to W$ from the metric space W, we use the notation $C_{\varepsilon}h$ to denote the family of curves $\gamma$ in $\Delta$ such that $h(\gamma) \in C_{\varepsilon}(W)$ . We need the following useful characterization of Sobolev mappings based on the limiting behavior of modulus of certain curve family, which generalizes [104, Theorem 3.10].

Theorem 6.15. A mapping $h : \tilde{\Omega}^f \to W$ belongs to the Sobolev space $N^{1,p}(\tilde{\Omega}^f, W)$ , 1 , if and only if $$\liminf_{\varepsilon \to 0} \varepsilon^p \operatorname{Mod}_p(\pi(\mathcal{C}_{\varepsilon}h)) < \infty.$$ Moreover, if this is the case, then the liminf on the left-hand side is an actual limit, and $$\int_{\Omega} \sup(|\nabla h|, y, \pi, \tilde{\Omega}^f) d\nu(y) = \lim_{\varepsilon \to 0} \varepsilon^p \operatorname{Mod}_p(\pi(\mathcal{C}_{\varepsilon}h)).$$

Corollary 6.16. Let p > 1 and let $\xi$ be a locally finite Borel regular measure on $\tilde{\Omega}^f$ . Suppose for every relatively compact open subset $\tilde{\Omega}_0^f \subset \tilde{\Omega}^f$ , $$\liminf_{\varepsilon \to 0} \varepsilon^p \operatorname{Mod}_p(\pi(\mathcal{C}_{\varepsilon} h|_{\tilde{\Omega}_0^f})) < \xi(\tilde{\Omega}_0^f).$$ Then $h \in N^{1,p}_{loc}(\tilde{\Omega}^f,W)$ and for $\lambda\text{-a.e. }z \in \tilde{\Omega}^f,$ $$|\nabla h|^p(z) \le \frac{d\xi}{d\lambda}(z).$$ The previous result, when applied to the special case $f = \iota$ , the inclusion mapping, reduces to the following result from [104, Theorem 3.10].

Corollary 6.17. Let p > 1, and $h: \Omega \to W$ , and let $\xi$ be a locally finite Borel regular measure on $\tilde{\Omega}$ . Suppose for every relatively compact open subset $\tilde{\Omega}_0 \subset \tilde{\Omega}$ , $$\liminf_{\varepsilon \to 0} \varepsilon^p \operatorname{Mod}_p(\mathcal{C}_{\varepsilon} h) < \xi(\tilde{\Omega}).$$ Then $h \in N_{loc}^{1,p}(\Omega, W)$ and for $\mu$ -a.e. $x \in \Omega$ , $$|\nabla h|^p(x) \le \frac{d\xi}{d\mu}(x).$$ As an immediate application of the above characterization of Sobolev mappings, we obtain the following characterization of analytic quasiregularity via the pullback factorization.

Proposition 6.18. Let $f: \tilde{\Omega} \to \Omega$ be a branched covering. Then f is analytically K-quasiregular with exponent Q if and only if $g: \tilde{\Omega} \to \Omega^f$ is analytically K-quasiconformal with exponent Q.

Proposition 6.19. Let $f: \tilde{\Omega} \to \Omega$ be a branched covering. Suppose $\Omega$ has c-bounded turning. Then for each $x_0 \in \tilde{\Omega}$ , we have $$\frac{1}{c}h_f(x_0) \le h_g(x_0) \le ch_f(x_0)$$ and $$\frac{1}{c}h_f^(x_0) \le h_g^(x_0) \le ch_f^*(x_0).$$ The analogous results hold with $h_f$ being replaced by $H_f$ , and $h_f$ being replaced by $H_f$ , respectively.

Theorem 6.20. If $\tilde{\Omega}$ and $\Omega$ are locally Ahlfors Q-regular, and $\Omega$ has c-bounded turning, then either of the following two conditions - i). $h_f(x) \leq h$ for all $x \in \tilde{\Omega}$ ; - ii). $h_f^*(x) \le h$ for all $x \in \tilde{\Omega}$ , implies that f is analytically $K_O$ -quasiregular with exponent Q and inverse analytically $K_I$ -quasiregular with exponent Q, with both constants $K_O$ and $K_I$ depending only on the constant of Ahlfors Q-regularity, and on c and h. Here, as was done in [4], we have assumed a stronger condition, that the dilatation is everywhere bounded, rather than simply everywhere finite and essentially bounded. We may drop this assumption (again, as in [4]), in the presence of a Loewner condition. In fact, even without the Loewner condition, it seems not be unnecessary to bound the dilatation everywhere, rather than essentially. However, this issue is rather technical, and so we eschew such considerations in this paper, as they would lead us too far astray. Note also that we did not impose the usual LLC condition on either domains, as we have done in [105], in the above theorem, though this condition is rather mild, and simplifies the exposition considerably. 6.3.1. Auxiliary results. The main result of this section is the following criterion for analytic quasiregularity, which generalizes [4, Theorem 1.1].

Theorem 6.21. Let $1 \leq p \leq Q$ and let $h: \Delta \to W$ be a homeomorphism between (pointwise) doubling metric measure spaces $(\Delta, d_{\Delta}, \sigma)$ and $(W, d_{W}, \tau)$ . Suppose there is a subset $E \subset \Delta$ such that for p-almost every curve $\gamma$ in $\Delta$ , $\mathcal{H}^{1}(h(\gamma \cap E)) = 0$ , and a function $\eta: \Delta \setminus E \to \mathbb{R}$ such that the following condition is satisfied: For every $v \in \Delta \setminus E$ and $\varepsilon > 0$ , there are neighborhoods $D_{v,\varepsilon}$ , $D'_{v,\varepsilon}$ and $D''_{v,\varepsilon}$ of v such that $D'_{v,\varepsilon} \cup D''_{v,\varepsilon} \subset D_{v,\varepsilon} \subset B(v,\varepsilon)$ , satisfying the inequalities $$\left(\frac{\operatorname{diam}\left(h(D_{v,\varepsilon})\right)}{\operatorname{diam}\left(D_{v,\varepsilon}\right)}\right)^{Q} \sigma(D'_{v,\varepsilon}) \leq \eta(v)\tau\left(h(D''_{v,\varepsilon})\right),$$ and $$\sigma(B(v, 10 \operatorname{diam}(D_{v,\varepsilon}))) \le C\sigma(D'_{v,\varepsilon}),$$ and satisfying the following property: For every subset $A \subset \Delta$ , and every set of indices $I \subset \Delta \times (0, \infty)$ such that $A \subset \bigcup_{\alpha \in I} D_{\alpha}$ , there is a countable subset $\{\alpha_i\} \subset I$ such that $A \subset \bigcup_{i=1}^{\infty} D_{\alpha_i}$ and whenever $i \neq j$ , $$D'_{\alpha_i} \cap D'_{\alpha_j} = \emptyset = D''_{\alpha_i} \cap D''_{\alpha_j}.$$ Then if p < Q and $\eta$ is essentially bounded, or p = Q and $\eta$ is bounded, then $h \in N^{1,p}(\Delta, W)$ , and the minimal p-weak upper gradient $|\nabla h|$ satisfies (6.5) $$\int_{\Delta} |\nabla h|^{Q} d\sigma \le C' \tau(\Delta),$$ where C' depends only on C and esssup $\eta$ We require the following generalization of a well-known covering lemma.

Lemma 6.22. Let $p \geq 1$ , let $\xi$ be a measure on $\tilde{\Omega}$ , and let $\{A_i\}$ and $\{B_i\}$ be sequences of $\xi$ -measurable sets and balls, respectively, such that for each $i \in \mathbb{N}$ , $A_i \subset B_i \subset \tilde{\Omega}$ , and $\xi(5B_i) \leq C\xi(A_i)$ , and let $\{c_i\}$ be a sequence of nonnegative integers. Then $$\int_{\tilde{\Omega}} \left( \sum_{i=1}^{\infty} c_i \chi_{B_i} \right)^p d\xi \le C_p \int_{\tilde{\Omega}} \left( \sum_{i=1}^{\infty} c_i \chi_{A_i} \right)^p d\xi,$$ where $C_p$ is a constant depending only on C and p.

Theorem 6.25. Suppose $\tilde{\Omega}$ has locally Q-bounded geometry, and $\Omega$ is locally Ahlfors Q-regular and locally LLC. Let $f: \tilde{\Omega} \to \Omega$ be an onto branched covering. Then the following conditions are quantitatively equivalent: - (1) f is metrically H-quasiregular; - (2) f is inverse metrically $H^*$ -quasiregular; - (3) f is weak metrically h-quasiregular with exponent Q; - (4) f is weak metrically $h^*$ -quasiregular with exponent Q; - (5) f is analytically K-quasiregular with exponent Q; - (6) f is geometrically K-quasiregular with exponent Q. Moreover, if these equivalent conditions are satisfied, then the inverse geometric and analytic definitions of quasiregularity hold with exponent Q as well. Bounded geometry in the target does not give us quite as much, but we still have the following inverse-to-forward result.

Theorem 6.26. If $\tilde{\Omega}$ is locally Ahlfors Q-regular and locally LLC, and $\Omega$ has locally Q-bounded geometry, then the inverse geometric or analytic definitions with exponent Q imply the corresponding forward with exponent Q, quantitatively. Combining Theorems 6.25 and 6.26 gives us our main result of this section.

Theorem 6.27. If both $\tilde{\Omega}$ and $\Omega$ have locally Q-bounded geometry, then all of the metric, geometric and analytic definitions with exponent Q are quantitatively equivalent. Theorem 6.27 can be viewed as a branched version of [41, Theorem 9.8], except that we do not yet have the notion of "branched quasisymmetric mappings". We will introduce such a natural class of branched quasisymmetric mappings in Section 6.7 and prove the quantitatively equivalence of (weak) metic quasiregularity and local branched quasisymmetry in Theorem 6.50. 6.4.1. Auxiliary results. We assume in thi section that $\tilde{\Omega}$ is locally Ahlfors Q-regular, locally LLC with constant $\lambda_{\tilde{\Omega}}$ , and that $\Omega$ has locally Q-bounded geometry. Note in particular that $\Omega$ has the local LLC property as well, quantitatively. Since $\tilde{\Omega}$ is locally compact, connected and locally connected, the local LLC property is equivalent to the local path LLC property. Since $\tilde{\Omega}$ is locally path LLC, it is not hard to see that $\tilde{\Omega}$ (and hence $\tilde{\Omega}^f$ as well) must be locally path connected, via the path-lifting property (cf. Lemma 5.4). We introduce one harmless topological assumption on $\tilde{\Omega}$ ; a non-quantitative version of the LLC property, which will always hold when we equip $\tilde{\Omega}$ with a metric giving it locally Q-bounded geometry. Namely, we assume that $\tilde{\Omega}$ is strongly locally connected, i.e., for every $x \in \tilde{\Omega}$ , and every open neighborhood $\tilde{\Omega}_0$ of x, there are open neighborhoods $\tilde{\Omega}_3 \subset \tilde{\Omega}_2 \subset \tilde{\Omega}_1 \subset \tilde{\Omega}_0$ of x such that every two points in $\tilde{\Omega}_1 \setminus \tilde{\Omega}_2$ can be joined in $\tilde{\Omega}_0 \setminus \tilde{\Omega}_3$ is connected. For the moment, it should be emphasized that we are not equipping $\tilde{\Omega}$ with a metric, and instead we will analyze the pullback metric space $\tilde{\Omega}^f$ under the assumption that $\Omega$ has locally Q-bounded geometry. We will use the notation $\mathbb{A}(x,r,R)$ to denote the annulus $B(x,R)\backslash \overline{B}(x,r)$ . We need the following result, which strengthens the strong local path LLC condition for $\tilde{\Omega}$ . Recall that the local path LLC condition requires that for each $x \in \Omega$ , there exists a radius $R_x > 0$ such that $B(x, R_x)$ is path LLC with constant $\lambda_{\tilde{\Omega}}$ .

Lemma 6.28. Let $n \in \mathbb{N}$ . Then there is a constant $\lambda_n$ , depending only on n and $\lambda_{\tilde{\Omega}}$ , such that for every $x \in \tilde{\Omega}$ , every $x_1, \ldots, x_n \in B(x, R_x/2)$ , every $r < R_x/2$ , and every pair of points $x', x'' \in B(x, R_x) \setminus \bigcup_{i=1}^n B(x_i, r)$ , there is a path joining x' to x'' in $B(x, \lambda_n R_x) \setminus \bigcup_{i=1}^n B(x_i, r/\lambda_n)$ . Proof. We argue by induction, as the case for n=1 follows easily from the local path LLC property of $\tilde{\Omega}$ . Suppose the result holds for n=k, and that we are given $r < R_x/2$ , k+1 points $x_1, \ldots, x_{k+1}$ , and points x' and x'' as in the statement of the lemma. By assumption, we may join x' and x'' with a path $\gamma \colon [0,1] \to B(x, \lambda_k R_x) \setminus \bigcup_{i=1}^k B(x_i, r/\lambda_k)$ . We may assume with no loss of generality that $d(x_{k+1}, x_i) \geq 2r/3\lambda_k$ for each $i=1,\ldots,k$ , since otherwise the conclusion holds with $\lambda_{k+1} = 3\lambda_k$ . Let $t_1, t_2 \in [0, 1]$ be the first and last values of t, respectively, for which $d(\gamma(t), x_{k+1}) \le r/(3\lambda_{\tilde{\Omega}}\lambda_k)$ , if any such values exist. Then we may modify $\gamma$ by replacing $\gamma|_{[t_1,t_2]}$ with a path joining $\gamma(t_1)$ to $\gamma(t_2)$ lying entirely in the annulus $\mathbb{A}(x_{k+1}, r/(3\lambda_{\tilde{\Omega}}^3\lambda_k), r/3\lambda_k)$ . Since $$\mathbb{A}(x_{k+1}, r/(3\lambda_{\tilde{\Omega}}^3\lambda_k), r/3\lambda_k) \subset B(x, \lambda_k R_x) \setminus \bigcup_{i=1}^k B(x_i, r/(3\lambda_{\tilde{\Omega}}^3\lambda_k)),$$ the conclusion holds for $\lambda_{k+1} = 3\lambda_{\tilde{O}}^3 \lambda_k$ . For a couple of standard modulus estimates we are going to present below, it is convenient to introduce the following concept, which essentials requires the projection mapping fullfills the assumption of Lemma 6.12.

Lemma 6.30. The radii R(z) in Proposition 4.2 may be chosen so that there are constants $C_1 > C_2 > 0$ , and $C_0 > 0$ , depending only on the data of $\Omega$ , such that for every $z \in \tilde{\Omega}^f$ , every C > 2, and every 2r < s < R(z), (6.8) $$\min\{C_1 \log(s/r)^{1-Q}, C_0\} \le \operatorname{Mod}_Q(\pi_z^{-1}(\mathcal{A}(z, r, s))) \le i_{\text{ess}}(z, \pi)C_2 \log(s/r)^{1-Q}.$$ Moreover, if $\pi$ has the good branching property, then the radii may chosen so that (6.9) $$\min\{C_1 \log(s/r)^{1-Q}, C_0\}i(z, \pi) \le \operatorname{Mod}_Q(\pi_z^{-1}(\mathcal{A}(z, r, s))) \le i_{ess}(z, \pi)C_2 \log(s/r)^{1-Q}.$$

Lemma 6.31. Let $z \in S$ be a $\pi$ -Lebesgue point of a Borel set $S \subset X$ and let $C \geq 2$ . Then $$\lim_{r\to 0} \operatorname{Mod}_{Q}(\pi_{z}^{-1}(\mathcal{A}(\pi(z), r, s)) \backslash \Gamma_{S}) = 0.$$

Lemma 6.32. The radii R(z) in Proposition 4.2, and the constants $C_1 > C_2 > 0$ from Lemma 6.30, may be chosen so that for $1 \le n \le N(\pi, \tilde{\Omega}_0^f)$ , $\lambda$ -a.e. $z \in D_n$ , and $2r < s < R(z)/\lambda_{\Omega}$ , $$(6.10) \quad \min\{C_1 \log(s/r)^{1-Q}, C_0\} \le \operatorname{Mod}_Q(\pi_z^{-1}(\mathcal{A}(\pi(z), r, s)) \cap \Gamma_{D_n}) \le C_2 \log(s/r)^{1-Q}.$$ Moreover, if $\pi$ has the good branching property, then the radii may chosen so that (6.11) $$\min\{C_1 \log(s/r)^{1-Q}, C_0\}i(z, \pi) \le \operatorname{Mod}_Q(\pi_z^{-1}(\mathcal{A}(\pi(z), r, s)) \cap \Gamma_{D_n}) \le C_2 \log(s/r)^{1-Q}.$$ We now come to our key geometric argument for this section. Recall that by the pullback factorization, $f = \pi \circ g$ , where $g \colon \tilde{\Omega} \to \tilde{\Omega}^f$ is a homeomorphism and $\pi \colon \tilde{\Omega}^f \to \Omega$ is a 1-BDD mapping.

Proposition 6.33. Let $\tilde{\Omega}$ be (pointwise) Ahlfors Q-regular and (locally) LLC, let $\Omega$ be a locally geodesic metric space with locally Q-bounded geometry, and let g satisfy the $K_I$ -inequality with $K_I = K$ with exponent Q. Then there is a constant $c = c(K, n) \geq 1$ , for each $n \in \mathbb{N}$ , depending only on K, n, and the data of $\Omega$ and $\tilde{\Omega}$ , such that R(z) may be chosen (not necessarily quantitatively) small enough, so that (6.12) $$\operatorname{diam}(g^{-1}(B(z,5r)))^{Q} \le c\mu(g^{-1}(B(z,r)))$$ for all $z \in \tilde{\Omega}^f$ with $i_{\text{ess}}(z, \pi) = n$ and r < R(z)/10.

Lemma 6.36. Suppose the branched covering $f: X \to Y$ satisfies both $K_O$ -inequality and $K_I$ -inequality with the same exponent Q. Then f is absolutely precontinous on Q-almost every curve $\gamma$ in X.

Lemma 6.38. If f : X → Y is absolutely precontinuous on γ : [a, b] → X, then f satisfies Condition N ⁻¹ on the curve γ, i.e. if E intersects γ on a set of positive H¹ -measure, then f(E) intersects f(γ) on a set of positive H¹ -measure. Proof. It suffices to show that if f(E) intersects f(γ) on a set of zero H¹ -measure, then E intersects γ on a set of zero H¹ -measure. Let β = f ◦ γ and E = γ(E0), where E⁰ ⊂ [a, b]. Let s^β : I → [0, l(β)] be the length function of β. Then γ = γ ^∗ ◦ s^β and γ ∗ : [0, l(β)] → X is absolutely continuous. Given any ε > 0, there exists some δ > 0 such that if {(bk, bk+1)} is a collection of disjoint intervals in [0, l(β)] with P k |bk+1 − bk| < δ, then P k d(γ ∗ (bk+1), γ^∗ (bk)) < ε. On the other hand, since H¹ (β(E0)) = 0, for any given δ > 0, there exists b^k = sβ(ak), where a^k ∈ E⁰ ⊂ I, such that P k |sβ(ak+1) − sβ(ak)| < δ. In particular, this implies that $$\mathcal{H}^1(\gamma(E_0)) \le \sum_k d(\gamma(a_{k+1}), \gamma(a_k)) = \sum_k d(\gamma^(b_{k+1}), \gamma^(b_k)) < \varepsilon.$$ We next point out the KO-inequality and K^I -inequality together (with the same exponent Q) imply both Condition N and Condition N ⁻¹ . In the lemma below, we assume that X has locally Q-bounded geometry, and Y is (locally) Ahlfors Q-regular and (locally) LLC.

Lemma 6.39. Suppose the branched covering f : X → Y satisfies both KO-inequality and K^I -inequality with the same exponent Q. Then f satisfies both Condition N and Condition N ⁻¹ .

Theorem 6.42. Let f : X → Y be a branched covering between two metric measure spaces. Suppose f satisfies the Poletsky's inequality with exponent Q and with constant K^I , i.e., for every curve family Γ in X, $$\operatorname{Mod}_Q(f(\Gamma)) \leq K_I \operatorname{Mod}_Q(\Gamma).$$ If, additionally, ν(f(B^f )) = 0, then f satisfies V¨ais¨al¨a's inequality with exponent Q and with the same constant K^I . Proof. Given Γ and Γ′ = f(Γ) as in the definition of V¨ais¨al¨a's inequality, it is clear that we only need to consider the rectifiable curves γ ′ ∈ Γ and hence we may assume each γ ′ : [0, l(γ ′ )] → Y is parametrized by arc-length. We may further assume that ModQ(Γ′ ) > 0, since otherwise there is nothing to prove. Denote by Γ⁰ the family of curves γ in X on which f fails to be absolutely precontinuous on a subcurve of γ. By Remark [6.37,](#page-61-2) ModQ(f(Γ0)) = 0. By Lemma [6.12](#page-43-1) and the above fact, we only need to show that for Q-almost every γ ′ ∈ Γ ′\f(Γ0), there are curves γ1, . . . , γ^m ∈ Γ\Γ⁰ and subcurves γ ′ 1 , . . . , γ′ ^m of γ ′ such that for each i = 1, . . . , m, γ ′ ⁱ = f(γi), and for almost every s ∈ [0, l(γ ′ )], γi(s) = γ^j (s) if and only if i = j. Towards a contradiction, suppose this is not true, then there exists a subfamily Γ′ ⁰ ⊂ Γ ′\f(Γ0) with ModQ(Γ′ 0 ) > 0 such that for each γ ′ ⊂ Γ ′ 0 , there are curves γ1, . . . , γ^m ∈ Γ\Γ⁰ and subcurves γ ′ 1 , . . . , γ′ ^m of γ ′ with γ ′ ⁱ = f(γi) for each i = 1, . . . , m, but for some i 6= j and some set E ⊂ [0, l(γ ′ )] of positive H¹ -measure, γi(s) = γ^j (s) for all s ∈ E. Since $\#\{k: \gamma_k(t) = x\} \le i(x, f), \ \gamma_i(E) \subset \mathcal{B}_f \ \text{and so } \gamma_i'(E) = f(\gamma_i(E)) \subset f(\mathcal{B}_f). \ \text{On the other}$ hand, since f is absolutely precontinuous on every curve $\gamma$ , we have by Lemma 6.38 that $$\mathcal{H}^1(\gamma'(E) \cap f(\mathcal{B}_f)) > 0$$ for every $\gamma' \in \Gamma'_0$ . In particular, this means that the family $\hat{\Gamma}$ of curves in Y that intersects $f(\mathcal{B}_f)$ with positive length is not null with respect to the Q-modulus, which contradicts with the fact that $\nu(f(\mathcal{B}_f)) = 0$ (cf. [42, Lemma 5.2.15]).

Theorem 6.43. Let $f: X \to Y$ be a homeomorphism between two Ahlfors Q-regular metric measure spaces. If $h_f(x) < \infty$ for all $x \in X$ , then f satisfies the weighted $K_O$ -inequality with exponent Q, i.e., $$\operatorname{Mod}_{Q,K_Q^{-1}}(\Gamma) \leq \operatorname{Mod}_Q(f(\Gamma)),$$ for some measurable function $K_O: X \to [1, \infty)$ . Moreover, if f satisfies Condition $N^{-1}$ , then there exists a measurable function $K_I: X \to [1, \infty)$ , such that f satisfies the weighted $K_I$ -inequality with exponent Q $$\operatorname{Mod}_Q(\Gamma) \leq \operatorname{Mod}_{Q,K_I}(f(\Gamma)).$$ For cleanness of our exposition, we did not restrict ourself to the most general situation. However, it is worth pointing out that similar results as in Theorem 6.43 hold if we replace the assumption $h_f < \infty$ by the symmetric one $h_f^* < \infty$ . There is nothing essentially new taking into account the asymmetry of our preceding arguments. Secondly, the assumption that f is a homeomorphism can be weakened as f is a branched covering, i.e., continuous, discrete and open mapping with locally bounded multiplicity. But some care need to be taken to conclude that f satisfies Condition N and/or Condition $N^{-1}$ on Q-almost every curves. This can be done by following the arguments from [105] and using the pullback factorization, but we do not repeat the arguments here and leave it as an exercise for those interested readers. 6.7. Branched quasisymmetric mappings. In the theory of metrically quasiconformal mappings, there is a proper subclass of mappings, termed quasisymmetric mappings, which carry stronger, global metric information, but less restrictive as those bi-Lipschitz mappings.

Proposition 6.47. A branched covering $f: X \to Y$ is branched $\eta$ -quasisymmetric if and only if $g: X \to X^f$ is generalized $\eta$ -quasisymmetric.

Proposition 6.48. Let X have $c_0$ -bounded turning and Y c-bounded turning, and let $f: X \to Y$ be a homeomorphism. Then f is generalized $\eta$ -quasisymmetric if and only if it is $\psi$ -quasisymmetric, quantitatively.

Theorem 6.50. Let f : X → Y be a weak metrically H-quasiregular mapping such that N = N(f, X) < ∞. Assume that both X and Y have locally Q-bounded geometry. Then f is locally η-branched quasisymmetric, quantitatively, with η depending only on H, N, and the data of X and Y . Proof. Let f = π ◦ g be the pullback factorization. Since f has bounded multiplicity, X^f has locally Q-bounded geometry. Since f : X → Y is weak metrically H-quasiregular, g : X → X^f is weak metrically H-quasiconformal, and by [\[4,](#page-88-3) Theorem 5.2 and Remark 5.3], g is locally η-quasisymmetric, quantitatively. The claim follows directly from Proposition [6.47](#page-69-0) and Proposition [6.48.](#page-69-1) Remark 6.51. In Theorem [6.50,](#page-70-0) the homeomorphism η depends, quantitatively on the multiplicity N. In general, one can not get rid of this dependence from the theorem, as the simple analytic function z 7→ z k indicated.

Theorem 7.4. Let $f:(M,g) \to (N,h)$ be a branched covering between two equiregular subRiemannian manifolds of homogeneous dimension $Q \geq 2$ and rank k. Then the following conditions are quantitatively equivalent: - 1) f is a metrically H-quasiregular mapping, - 2) f is a weak metrically H-quasiregular mapping, - 3) f is a horizontally $\widehat{H}$ -quasiregular mapping with exponent Q, - 4) f is an analytically K-quasiregular mapping with exponent Q, - 5) f is a geometrically K-quasiregular mapping with exponent Q. Moreover, we have the following precise dependences on the quasiregularity constants $H, \widehat{H}$ , and K: - If f is weak metrically H-quasiregular, then it is analytically K-quasiregular with $K = H^{Q-1}$ and horizontally $\widehat{H}$ -quasiregular with $\widehat{H} = H^{k-1}$ . - If f is analytically K-quasiregular, then it is metrically H-quasiregular with H = K and horizontally $\widehat{H}$ -quasiregular with $\widehat{H} = K$ . - If f is horizontally $\widehat{H}$ -quasiregular, then f is analytically K-quasiregular with $K = \widehat{H}^{Q-1}$ and metrically H-quasiregular with $H = \widehat{H}$ . Based on Theorem 7.4, we will simply say that $f: M \to N$ is a K-quasiregular mapping if it is K-quasiregular according to one of five definitions in Theorem 7.4. We also refer the interested readers to [29] for more analytic properties of quasiregular mappings in the subRiemannian manifolds. As a particular application of our general theory of quasiregular mappings studied in the previous section, we obtain the important Väisälä's inequality when the subRiemannian manifolds are Ahlfors regular.

Corollary 7.5. Let $f: M \to N$ be a K-quasiregular mapping between two Ahlfors Qregular, $Q \geq 2$ , equiregular subRiemannian manifolds. Then f satisfies the Väisälä's\ninequality with constant $K_I$ , where $K_I$ depends only on K and the Ahlfors regularity constants of M and N. Proof. By either [30, Corollary 1.2] or [29, Theorem B], we have $\mathcal{H}^Q(f(\mathcal{B}_f)) = 0$ whenever $f \colon M \to N$ is metrically K-quasiregular. By the proof of Theorem 6.20 and Remark 6.24, we know that f satisfies the Poletsky's inequality with some constant $K'_I$ that depends quantitatively on K and on the Ahlfors regularity constants of the spaces. Finally, the claim follows from Theorem 6.42. We do not know whether an Ahlfors Q-regular equiregular subRiemannian manifold M necessarily has locally Q-bounded geometry. If so, Corollary 7.5 would follow directly from Theorem 6.27. It is also clear from the above proof that Corollary 7.5 remains valid if both M and N are uniformly locally Ahlfors Q-regular, i.e., there exists a positive constant $C_d$ such that for each point x, there is some positive radius $r_x$ making the measure Vol Ahlfors Q-regular with constant $C_d$ for the metric measure space $(B(x, r_x), d, \text{Vol})$ . In other words, we require the Ahlfors regularity constant is uniform, but allowing the radius vary at each point. As an obvious consequence of Theorem [7.4](#page-73-0) and Corollary [7.5,](#page-73-1) we point out that under the same assumptions as in Corollary [7.5,](#page-73-1) the class of quasiconformal mappings form a group. Namely, if f : M → N is a K-quasiconformal mapping, then f −1 : N → M is K′ -quasiconformal, with K′ depending only on K and the Ahlfors regularity constants of M and N.

Theorem 8.1. Let f : X → Y be a BDD branched covering, with N = N(f, X) < ∞, and suppose that Y is doubling and that X and Y have bounded turning. Then there is a bi-Lipschitz embedding of X into Y × R ^cd(N−1), where c^d depends only on the data of Y . Recall that a metric space X is doubling with constant M, M ≥ 1 an integer, if for each ball B(x, r) in X, every r/2-separated subset of B(x, r) has at most M points. As a consequence of Theorem [8.1,](#page-75-3) we obtain that the domain of a BDD branched covering inherits bi-Lipschitz embeddability into Euclidean space from the image. Taking into account of Proposition [5.14,](#page-38-0) we deduce the following corollary.

Corollary 8.2. Let f : X → Y be a BDD branched covering, with N = N(f, X) < ∞, where X and Y have bounded turning and Y embeds bi-Lipschitzly into some Euclidean space. Then X may also be embedded bi-Lipschitzly into some (possibly larger) Euclidean space. In particular, every BLD covering of a locally compact quasiconvex metric space over R ⁿ with finite maximal multiplicity embeds bi-Lipschitzly into some Euclidean space. In general, we are restricting our attention as much as possible to the global statements to simplify the exposition; the numerous formulations and proofs of similar local statements are left as routine exercises for the interested readers. We do, however, wish to highlight a few consequences arising from localization. First, by localizing Corollary [8.2,](#page-75-2) we obtain that every locally BLD-Euclidean metric space of dimension n admits local bilipschitz embeddings into Euclidean spaces, with the dimension of the local embeddings depending only on n and the local degree of the BLD covering. This answers in the affirmative an open question of Heinonen and Rickman [\[43,](#page-89-4) Remark 6.32 (a)]. Of course, under such localized assumptions, the dimensions of the Euclidean spaces receiving the neighborhoods of X might very well be unbounded. As a result, we cannot hope to obtain, from our methods, a single globally defined, locally bijective mapping from X into a Euclidean space. If, on the other hand, conditions are "uniformly local", in the sense that their associated constants are still defined globally, then we are able to obtain such a mapping.

Theorem 8.3. Let f : X → Y be a locally BDD branched covering, and suppose that X and Y have uniformly locally bounded turning, that Y is uniformly locally doubling, and that f has uniformly locally bounded multiplicity M < ∞. Then there is a globally defined, uniformly locally bi-Lipschitz mapping φ: X → Y × R ^cd(M−1), where c^d depends only on the data of Y . By using the typical gluing argument, together with the Nashing embedding theorem, it is plausible that one can update Theorem [8.3](#page-76-1) so that φ is globally injective, and hence is bi-Lipschitz with respect to the length metric. For simplicity of our exposition, we do not get involved with this technical issue, but leave it to those interested readers. Another observation we would like to point out is that as a consequence of Theorem [8.1,](#page-75-3) we have for each K-quasiregular mapping f : R ⁿ → R ⁿ a canonical factorization f = π◦g, where g : R ⁿ → X^f is K-quasiconformal and π : X^f → R ⁿ with the metric space X^f being a generalized n-manifold of type A (in the sense of Heinonen-Rickman [\[43\]](#page-89-4)). This provides a ncie motivation of the study of BLD mappings from generalized n-manifolds of type A to R ⁿ and quasiconformal mappings from R n to generalized n-manifolds of type A. 8.2. Proof of the main results. We first make a simple reduction, which is not strictly speaking necessary, but will allow us a somewhat cleaner exposition. Since X and Y have bounded turning, and f is BDD, we have by Remark 3.7 that Y is bi-Lipschitz equivalent to ι ^∗Y and X is bi-Lipschitz equivalent to f ^∗Y = f ∗ ι ^∗Y , and so we may assume with no loss of generality that X and Y have 1-bounded turning, and f is 1-BDD. Notice that in the proof of Theorem [8.1](#page-75-3) given below, c^d will depend explicitly only on the doubling constant, but because of our reduction, it implicitly depends on the constant of bounded turning in Y as well. Motivated by the fact that the theorem is trivial when f is one-to-one, our proof of Theorem [8.1](#page-75-3) begins with the following simple lemma, which makes quantitative the fact that the lack of injectivity is our only obstacle.

Lemma 8.4. Let f : X → Y satisfy the hypotheses of Theorem [8.1.](#page-75-3) Suppose that for some ε > 0, there is an L-Lipschitz mapping φ: X → W into some metric space W such that for all x, x′ ∈ X such that f(x) = f(x ′ ), we have εd(x, x′ ) ≤ d(φ(x), φ(x ′ )). Then the mapping $$\psi = f \times \phi \colon X \to Y \times W$$ is a bi-Lipschitz embedding. Proof. That ψ is Lipschitz is clear. For the reverse inequality, let 0 < δ < ^ε L+ε , so that ε(1 − δ) − Lδ > 0, and suppose first that x1, x² ∈ X, with d(y1, y2) ≤ δd(x1, x2). Here we denote $y_i = f(x_i)$ , $w_i = \phi(z_i)$ . Since f is 1-BDD and X, Y have 1-bounded turning, there is some $x'_1 \in f^{-1}(y_1)$ such that $$d(x_1', x_2) = d(y_1, y_2) \le \delta d(x_1, x_2),$$ whereby $$(1-\delta)d(x_1,x_2) \le d(x_1,x_2) - d(x_1',x_2) \le d(x_1,x_1').$$ Then we have $$(1 - \delta)d(x_1, x_2) \le d(x_1, x_1') \le \frac{d(w_1, w_1')}{\varepsilon} \le \frac{d(w_1, w_2) + d(w_2, w_1')}{\varepsilon}$$ $$\le \frac{d(w_1, w_2) + Ld(x_2, x_1')}{\varepsilon} \le \frac{d(w_1, w_2) + L\delta d(x_1, x_2)}{\varepsilon},$$ and so $$(\varepsilon(1-\delta) - L\delta)d(x_1, x_2) \le d(w_1, w_2).$$ Thus, for every $x_1, x_2 \in X$ , we have $$\min \left\{ \varepsilon(1-\delta) - L\delta, \delta \right\} d(x_1, x_2) \le \max \left\{ d(w_1, w_2), d(y_1, y_2) \right\}.$$ Remark 8.5. It is clear from the proof that the bi-Lipschitz constant associated to the embedding depends quantitatively only on $\varepsilon$ , L and the bounded turning constants of X and Y. Proofs of Theorem 8.1 and Theorem 8.3. For each $y \in Y$ , and each k = 1, ..., N - 1, let $R^k(y)$ be the smallest number such that $f^{-1}(B(y, 5R^k(y)))$ consists of at most k (connected) components. We have that $R^k(y) > 0$ if and only if N(y, f) > k, that $$0 = R^{N}(y) = R^{N(y,f)}(y) < R^{N(y,f)-1}(y) \le \dots \le R^{1}(y),$$ and that for all $x_1 \neq x_2 \in f^{-1}(y)$ , there is some $k, 1 \leq k \leq N(y, f) - 1$ , such that (8.1) $$\frac{1}{2}d(x_1, x_2) \le 5R^k(y) \le d(x_1, x_2);$$ the last statement follows from Lemma 5.7. Fix k, and let $B_y^k = B(y, R^k(y))$ for each $y \in Y$ , so that $B_y^k$ is nonempty if and only if N(y, f) > k. Also, let $$Y^k = \big\{ y \in Y : N(y,f) > k \big\} = \big\{ y \in Y : R^k(y) > 0 \big\}.$$ Since balls are connected in Y, and $f^{-1}(B(y,r))$ has more than k components for each $r < 5R^k(y)$ , Lemma 5.4 implies that each $y' \in 5B_y^k$ has more than k preimages, hence $5B_y^k \subset Y^k$ . (Observe this technically holds for all $y \in Y$ , since the open ball $5B_y^k$ is empty for $y \notin Y^k$ .) Thus we actually have $Y^k = \bigcup_{y \in Y} 5B_y^k = \bigcup_{y \in Y} B_y^k$ . Moreover, for every $y, y' \in Y$ , the inclusion $B(y, 5R^k(y)) \subset B(y', 5R^k(y) + d(y, y'))$ implies $$5R^{k}(y') < 5R^{k}(y) + d(y, y'),$$ so that $R^k$ is $\frac{1}{5}$ -Lipschitz in y. This in turn implies that for every $y, y' \in Y$ such that $B^k_y \cap B^k_{y'} \neq \emptyset$ , we have $$R^{k}(y') - R^{k}(y) \le \frac{1}{5}d(y', y) \le \frac{1}{5}(R^{k}(y') + R^{k}(y)),$$ so that (8.3) $$\frac{2}{3}R^{k}(y) \le R^{k}(y') \le \frac{3}{2}R^{k}(y),$$ and similarly, for every $y, y' \in Y$ such that $2B_y^k \cap 2B_{y'}^k \neq \emptyset$ , (8.4) $$\frac{3}{7}R^{k}(y) \le R^{k}(y') \le \frac{7}{3}R^{k}(y).$$ We next choose a maximal subset $T^k \subset Y^k$ such that for each $y, y' \in T^k$ , $d(y, y') \ge \frac{1}{2}R^k(y)$ . By the inequality (8.3), $Y^k = \bigcup_{y \in T^k} B^k_y$ and by inequality (8.4), for each $y \in T^k$ there are at most $c_d - 1$ other points $y \in T_k$ such that $2B^k_y \cap 2B^k_{y'} \ne \emptyset$ , where $c_d$ depends only on the doubling constant of Y. We may therefore choose $c_d$ disjoint subsets $T^k_j \subset T^k$ , $j = 1, \ldots, c_d$ , such that $T^k = \bigcup_{j=1}^{c_d} T^k_j$ , and for each $y, y' \in T^k_j$ , $y \ne y'$ , we have $2B^k_y \cap 2B^k_{y'} = \emptyset$ . Now let $X^k = f^{-1}(Y^k)$ . For each $x \in X^k$ , let $U_x^k = U(x, f, R^k(f(x)))$ , and for $\lambda \geq 0$ , let $\lambda U_x^k = U(x, f, \lambda R^k(f(x)))$ . We also let $S^k = f^{-1}(T^k)$ , and for each $j = 1, \ldots, c_d$ , let $S_j^k = f^{-1}(T_j^k)$ . We then have $X^k = \bigcup_{x \in S^k} U_x^k$ , and for each $x, x' \in S_j^k$ , the sets $2U_x^k$ are either disjoint or equal, the latter holding if and only if f(x) = f(x'). It follows that we may choose a function $\eta_j^k \colon S_j^k \to \{1, \ldots, N\}$ such that for each $y \in Y^k$ and each $x_1, x_2 \in f^{-1}(y), \eta_j^k(x_1) = \eta_j^k(x_2)$ if and only if $2U_{x_1}^k = 2U_{x_2}^k$ . Note, as well, that since a continuum contained in $f^{-1}(B_{f(x)}^k)$ must lie completely inside $2U_x^k$ , and X has 1 bounded turning, we have $d(X \setminus 2U_x^k, U_x^k) \geq R^k(f(x))$ , for all $x \in X^k$ . Now let $\phi_i^k \colon X \to \mathbb{R}$ be defined by $$\phi_j^k(x) = \sum_{x' \in S_j^k} \eta_j^k(x') \min \left\{ d(x, X \setminus 2U_{x'}^k), R^k(f(x')) \right\},$$ and let $\phi: X \to \mathbb{R}^{c_d(N-1)}$ be given by $$\phi = \phi_1^1 \times \dots \times \phi_i^k \times \dots \times \phi_{c_d}^{N-1}.$$ To complete the proof, it suffices to show that $\phi$ satisfies the hypotheses of Lemma 8.4. To this end, we first note that each $\phi_j^k$ is clearly N-Lipschitz, and satisfies $\eta(x')R^k(f(x'))$ whenever $x \in U_x^k$ . It follows that $\phi$ itself is Lipschitz. On the other hand, if $x_1 = x_2$ , and $y = f(x_1) = f(x_2)$ , then let k be chosen so that inequality (8.1) holds. A fortiori, $y \in Y_k$ , so $y \in B_y^k$ for some $y \in T_j^k$ , some j. By Lemma (5.4), we have $x_1 \in U_x^k$ , $x_2 \in U_x^k$ for some $x_1', x_2' \in f^{-1}(y') \subset S_j^k$ . In fact, we may take $x_i \in B_{x_i'}^k := B(x_i', R_k(y))$ , since by lemma 5.7, $U_x^k$ is a union of such balls. Thus, applying inequalities (8.1) and (8.2), we obtain $$d(x'_1, x'_2) \ge d(x_1, x_2) - d(x'_1, x_1) - d(x_2, x'_2) > 5R^k(y) - 2R^k(y')$$ $$\ge 5R^k(y') - d(y, y') - 2R^k(y) \ge 2R^k(y'),$$ and so by Lemma 5.7, $U_{x_1'}^k \neq U_{x_2'}^k$ , so that $\eta_j^k(x_1') \neq \eta_j^k(x_2')$ . Moreover, since $x_i \in U_{x_i'}$ , we have $\phi_j^k(x_i) = \eta_j^k(x_i')R^k(y')$ , i = 1, 2. We therefore have, again, by inequalities (8.1) and (8.2), that $$d(x_1, x_2) \le 10R^k(y) \le 10R^k(y') + 2d(y, y') \le 12R^k(y')$$ $$\le 12|\eta_j^k(x_1') - \eta_j^k(x_2')|R^k(y') = 12|\phi_j^k(x_1) - \phi_j^k(x_2)| \le 12|\phi(x_1) - \phi(x_2)|.$$ Thus $\phi$ satisfies the hypotheses of Lemma (8.4), as desired. 8.3. Some natural examples and counter-examples. In this section, we use the pullback metric to construct branched coverings between nice metric spaces so that the image of the branch set under the branched covering will have positive Lebesgue measure. Follow [43, 45], we say that an oriented generalized n-manifold X is locally BLDEuclidean of dimension n if each point $x \in X$ has a neighborhood $U_x$ together with a Lipschitz BLD mapping $f: U_x \to \mathbb{R}^n$ . By [45, Proposition 6.31], a locally BLD-Euclidean space X of dimension n is a Lipschitz n-manifold outside a singular set of topological dimension at most n-2. Moreover, X is locally Ahlfors n-regular, n-rectifiable and has locally finite $\mathcal{H}^n$ -measure. Our main result of this section reads as follows.

Theorem 8.6. Let Z be a topological (or generalized) n-manifold, and $g: Z \to \mathbb{M}$ a branched covering onto a Riemannian n-manifold $\mathbb{M}$ , such that $N(g,Z) < \infty$ and that $\mathcal{L}^n(g(\mathcal{B}_g)) > 0$ . Then $\mathbb{M}^g$ is a locally BLD-Euclidean metric space of dimension n that is neither locally linearly locally contractible nor locally metrically orientable. In particular, $\mathbb{M}^g$ is n-rectifiable, locally Ahlfors n-regular, locally geodesic, locally satisfies a (1,1)-Poincaré inequality, and has local bi-Lipschitz embeddings into some Euclidean space, but is not locally quasiconformally equivalent to any neighborhood of $\mathbb{R}^n$ . It is not hard to construct topological branched coverings for which the image of the branch set has positive measure. As a result, Theorem 8.6 provides a rather rich source of examples and counterexamples. For example, a conjecture of Heinonen and Rickman [43, Remark 6.32 (b)] is that the branch set of a Lipschitz BLD map $f: X \to \mathbb{R}^n$ from a generalized n-manifold into Euclidean space has Hausdorff n-measure 0, provided that X has local bi-Lipschitz embeddings into some larger Euclidean space. As a corollary to Theorem 8.6 (and invoking also Corollary 8.2), we can show that this conjecture is false, and that there are in fact counterexamples of degree 2 in all dimensions, even under the restrictions that X is homeomorphic to $\mathbb{R}^n$ and the bilipschitz embedding is global.

Corollary 8.7. For each $n \geq 3$ , there is a subspace $X \subset \mathbb{R}^N$ for some $N \geq n$ , homeomorphic to $\mathbb{R}^n$ , and a 1-BLD branched covering $f: X \to \mathbb{R}^n$ of degree 2 such that $\mathcal{L}^n(\mathcal{B}_f) > 0$ . Heinonen and Semmes also asked [\[44,](#page-89-23) Question 33] if every closed topological 4 manifold admits a metric so that it is Ahlfors 4-regular and locally linearly locally contractible. (The answer to this question is "yes" for manifolds of dimension n 6= 4, by the work of Sullivan [\[90\]](#page-91-12) showing that every such manifold has a Lipschitz structure.) A motivation for this is that an affirmative answer would imply, by the work of Semmes [\[84\]](#page-91-13), that every closed topological 4-manifold admits an n-regular, n-rectifiable metric satisfying a (1, 1)-Poincar´e inequality. The pullback construction gives another possible avenue to the question of whether 4-manifolds can be metrized to admit Poincar´e inequalities. Heinonen and Semmes [\[44,](#page-89-23) Question 31] asks if every closed topological 4-manifold is a branched cover of S 4 . If the answer to this question is "yes", then by Theorem [8.6,](#page-79-1) we may indeed give each 4-manifold such a metric. Proof of Theorem [8.6.](#page-79-1) Let π : g ^∗M → M be the projection (from the pullback factorization). That g ^∗M is a locally BLD-Euclidean space follows directly from Lemma [5.7.](#page-33-0) Note that if g ^∗M is either locally linearly locally contractible or locally metrically orientable, then it would follow from [\[43,](#page-89-4) Theorem 6.4] that L n (g(Bg)) = L n (π(Bπ)) = 0. Thus g ^∗M is neither locally linearly locally contractible nor locally metrically orientable. That g ^∗M is locally n-rectifiable, locally Ahlfors n-regular, locally geodesic also follows immediately from Lemma [5.7,](#page-33-0) Lemma [5.8,](#page-34-0) and the local geometry of the Riemannian n-manifolds M. The fact that g ^∗M supports a (1, 1)-Poincar´e inequality follows directly from [\[43,](#page-89-4) Theorem 9.8]. The last assertion is a direct consequence of Corollary [8.2.](#page-75-2) Proof of Corollary [8.7.](#page-79-2) Choose a topological branched covering g : R ⁿ → R ⁿ with degree 2 (for instance the standard winding mapping; e.g. [\[80,](#page-91-0) Example I 3.1]) so that L n (g(Bg)) > 0 and then invoke Theorem [8.6.](#page-79-1) 8.4. Geometric parametrization of metric spaces. In the field of analysis and geometry on metric spaces, one of the celebrated open problems is to find good geometric parametrization of certain classes of metric spaces (see e.g. [\[36,](#page-89-21) Section 16.5]). For the bi-Lipschitz parametrization, it was initiated earliest by an observation due to Siebenmann and Sullivan [\[89\]](#page-91-14). Remarkable positive parametrization results were achieved by Toro [\[92,](#page-91-15) [93\]](#page-91-16). There are also several highly non-trivial results, both positive and negative, were obtained by Semmes [\[82,](#page-91-17) [86,](#page-91-10) [85\]](#page-91-9), David–Semmes [\[19,](#page-88-14) [20\]](#page-88-15), Bonk– Lang [\[14\]](#page-88-16), Bonk–Heinonen–Saksman [\[11\]](#page-88-17), and Heinonen–Rickman [\[43\]](#page-89-4). Inspired by Sullivan's work [\[90,](#page-91-12) [91\]](#page-91-18), a simple geometric condition, that is sufficient for a space to admit local bi-Lipschitz parametrization by the Euclidean spaces, was provided in the remarkable works of Heinonen–Sullivan [\[45\]](#page-89-8) and Heinonen–Keith [\[38\]](#page-89-5). Instead of the bi-Lipschitz parametrization, people also ask for weaker geometric parametrizations, e.g. quasiconformal or quasisymmetric parametrization. The research along this direction often seeks for a version of the classical uniformization theorem for a certain class of two-dimensional metric spaces. Uniformization problems concerning quasiconformal and quasisymmetric mappings have received considerable attention in recent years, and they have found significant applications in geometry, complex dynamics, geometric topology and geometric measure theory, among other areas. In particular, several problems in the theory of hyperbolic groups can be interpreted as uniformization problems concerning boundaries of the groups in question; see for instance [\[8,](#page-88-18) [9,](#page-88-19) [13,](#page-88-5) [15,](#page-88-20) [16,](#page-88-21) [52\]](#page-89-24) and also [\[94,](#page-91-7) [17,](#page-88-22) [3,](#page-88-23) [67,](#page-90-14) [68\]](#page-90-16). We would like to mention the remarkable result of Bonk–Kleiner [\[12\]](#page-88-4), where it has been shown that for an Ahlfors 2-regular topological sphere, it is quasisymmetric to the standard sphere S 2 if and only if it is linearly locally contractible (LLC\). LLC\ is a geometric condition that rules out cusp-like spaces. This beautiful result was later extended in several consequent works [\[66,](#page-90-17) [102,](#page-91-19) [103\]](#page-91-20). Very recently, in a celebrated result of Rajala [\[75\]](#page-90-9), a quasiconformal analogy of the Bonk–Kleiner result was obtained via a geometric approach. In higher dimensions, the uniformization problem does not have a satisfactory answer even for very nice metric spaces. Examples by Semmes [\[85\]](#page-91-9) show that the result of Bonk– Kleiner mentioned above does not generalize to dimension 3. Heinonen and Wu [\[46\]](#page-89-25), Pankka and Wu [\[71\]](#page-90-18) and Pankka and Vellies [\[72\]](#page-90-19) gave further examples of geometrically nice spaces without quasisymmetric parametrizations. - 9. Characterizations of BLD mappings in metric spaces with bounded geometry - 9.1. Background and formulation. In this section, we give another application of the pullback factorization and the theory of quasiregular mapping that we have developed in Section [6.](#page-39-0) BLD mappings in Euclidean spaces were first introduced by Martio and V¨ais¨al¨a [\[65\]](#page-90-20) in their study of second-order elliptic operators. They established many interesting analytic and geometric properties of BLD mappings in R ⁿ via the theory of quasiregular mappings. In particular, they obtained the following quantitative analytic characterization of BLD mappings: a continuous mapping f : Ω → R n , n ≥ 2, is BLD if and only if f is locally uniformly Lipschitz and the Jacobian determinant J^f = det Df is positive and uniformly bounded away from zero almost everywhere in Ω. This description of BLD mappings in Euclidean spaces does not include the assumption that the mapping is discrete and open, nor that it is sense-preserving. In fact, mappings satisfying the latter analytic conditions as above form a strict subclass of quasiregular mappings. A deep theorem of Reshetnyak [\[80\]](#page-91-0) implies that (non-constant) quasiregular mappings are both discrete and open, and consequently they are sense-preserving as well. In [\[45\]](#page-89-8), Heinonen and Sullivan successfully generalized Reshetnyak's theorem to quasiregular mappings from generalized n-manifolds of type A into R n . As a typical application of this result, Heinonen and Rickman [\[43,](#page-89-4) Theorem 6.18] have obtained a similar analytic characterization of BLD mappings in the setting of mappings from generalized n-manifolds of type A to R n . Our main result of this section is the following quantitative characterizations of BLD mappings in Ahlfors Q-regular Q-Loewner spaces. Throughout the entire section, Q will be a real number strictly larger than one and n ≥ 2 an integer.

Theorem 9.1. Let f : X → Y be a branched covering between two Ahlfors Q-regular Q-Loewner spaces. Then the following statements are equivalent: - 1). f is L-BLD; - 2). For each x ∈ X, there exists r^x > 0 such that $$\frac{d(x,y)}{c} \le d(f(x), f(y)) \le cd(x,y)$$ for all y ∈ B(x, rx); - 3). L^f (x) ≤ c and l^f (x) ≥ 1 c for each x ∈ X; - 4). f is metrically H-quasiregular, locally M-Lipschitz, and J^f (x) ≥ c for a.e. x ∈ X. Moreover, all the constants involved depend quantitatively only on each other and on the data associated to X and Y . Recall that for a mapping f : X → Y , L^f and l^f are defined as $$L_f(x) = \limsup_{y \to x} \frac{d(f(x), f(y))}{d(x, y)} \quad \text{and} \quad l_f(x) = \liminf_{y \to x} \frac{d(f(x), f(y))}{d(x, y)}.$$ The assumption that f is K-quasiregular in Theorem 9.1 4) can be dropped, since it is implied by the other two conditions; see Lemma 9.5 below. We prefer the current formulation simply because we expect that Theorem 9.1 would hold in a wider class of metric spaces, where the (metric) quasiregularity does not necessarily follow from the locally Lipschitz regularity and the lower positive bounds on the (volume) Jacobian. As commented in the beginning of this section, Theorem 9.1 was first proved in the Euclidean spaces by Martio and Väisälä [65, Theorem 2.16], and Later, generalized by Heinonen and Rickman [43, Theorem 6.18], to mappings from generalized n-manifolds of type A into the Euclidean space $\mathbb{R}^n$ . The proof of Martio and Väisälä depends not only on the geometry of Euclidean spaces, but also on the differentiable structure of Euclidean spaces, thus it cannot be generalized to our setting. Heinonen and Rickman were able to give a proof independent of the differentiable structure of Euclidean spaces, but their proof still depends heavily on the geometry of Euclidean spaces (as the target space). It seems for us not so easy to adjust their proof to the setting of mappings between two generalized n-manifolds of type A. The new approach we have developed here is to use the pullback factorization from Section 5.2 to factorize f as $f = \pi \circ g$ and then transfer the information of f to its lift mapping $g \colon X \to X^f$ . Then using the techniques from Section 6 and the basic properties of pullback factorizations to show that g is bi-Lipschitz, quantitatively. The main obstacle here is to update the $\mu$ -a.e. pointwise information of f in Theorem 9.1 4) to the everywhere defined BLD condition in Theorem 9.1 1). At the level of f, this is not an easy task as already observed in [43, Lemma 6.19, Lemma 6.23 and Proof of Theorem 6.18]. Somewhat surprisingly, this is quite easy to handle, via a simple lemma (see Lemma 9.3 below), at the level of the lifting mapping g. Very recently, Luisto [59] has shown that the equivalences of 1)–3) hold in very general metric spaces. Moreover, for a continuous discrete mapping $f: X \to Y$ between length spaces, it has been shown that f is L-BLD if and only it is L-Lipschitz Quotient (L-LQ for short), i.e., for each $x \in X$ and r > 0, (9.1) $$B(f(x), L^{-1}r) \subset f(B(x,r)) \subset B(f(x), Lr).$$ Based on the afore-mentioned characterizations, Luisto has obtained an interesting convergence result that generalizes earlier works of Martio-Väisälä [65, Theorem 4.7], Heinonen-Keith [38, Lemma 6.2] and Luisto [58, Corollary 1.3]. Using the recent result on the size of the branch set of a quasiregular mapping obtained by the authors [30], we are able to improve on Luisto's result into the following form.

Theorem 9.2. Let X and Y be two Ahlfors Q-regular generalized n-manifolds. Assume additionally that X is locally linearly locally contractible and locally quasiconvex and that Y is locally quasiconvex. Let $\{f_i\}_{i\in\mathbb{N}}$ be a sequence of L-BLD mappings converging locally uniformly to a continuous mapping $f: X \to Y$ . Then f is L-BLD. Recall that X is locally linearly locally contractible if for each $x \in X$ , there exists a neighborhood $U_x$ of x such that $U_x$ is $c_x$ -linearly locally contractible, i.e., each ball $B(y,r) \subset U_x$ is contractible in $B(y,c_xr)$ . The assumptions that X and Y are locally quasiconvex can be further relaxed; see Remark 9.7 below. Our interest in Theorem 9.2 lies in seeking for an easy proof of the geometric porosity for the branch set of a BLD mapping in metric spaces via a blow up argument. We will not enter any technical details, since it clearly beyonds the scope of the current paper, but we will address this issue in a sequential paper. 9.2. Auxiliary results. The following lemma is certainly well-known to experts. However, we present a simple proof here due to lack of a precise reference.

Lemma 9.3. Let $h: X \to Y$ be a continuous $N_{loc}^{1,Q}(X,Y)$ -mapping between two Ahlfors Q-regular Q-Loewner spaces such that the constant function c is a Q-weak upper gradient of h. Then h is C-Lipschitz, quantitatively.

Theorem 9.4. Let $f: X \to Y$ be an L-BLD mapping between two Ahlfors Q-regular generalized n-manifolds with $\mathscr{H}^Q(f(\mathcal{B}_f)) = 0$ . Assume that X is $c_X$ -quasiconvex and that Y is $c_Y$ -quasiconvex. If $x_0 \in X$ , r > 0, and if $\lambda > 1$ such that the ball $B(x_0, \lambda r)$ has compact closure in X, then $$N(y, f, B(x_0, r)) \le \left(Lc_X\right)^Q \frac{\mathcal{H}^Q(B(x_0, \lambda r))}{\mathcal{H}^Q(B(y, (\lambda - 1)r/Lc_Y))}$$ for all $y \in Y$ . The following lemma is an easy consequence of Theorem 6.27.

Lemma 9.5. Let $f: X \to Y$ be a branched covering between two Ahlfors Q-regular Q-Loewner metric spaces. If f is locally M-Lipschitz and if $J_f > c$ a.e. in X, then f is metrically H-quasiregular, quantitatively.

Lemma 9.6. Let X and Y be two locally quasiconvex metric spaces. Then $f: X \to Y$ is L-BLD if and only if $g: \hat{X} \to \hat{Y}$ is L-BLD, where g = f on the set X.

Definition 1.1. A branched covering f : X → Y between two metric measure spaces is termed metrically H-quasiregular if the linear dilatation function H^f is finite everywhere and essentially bounded from above by H. If f : X → Y , in Definition [1.1,](#page-2-0) is additionally assumed to be a homeomorphism, then f is called metrically H-quasiconformal. We will call f a metrically quasiregular/quasiconformal mapping if it is metrically H-quasiregular/quasiconformal for some H ∈ [1, ∞). Note that our definition of metric quasiconformality allows an exceptional set for the boundedness of the linear dilatation, which is in general weaker than the more commonly used definition in literature, where the linear dilatation is required to be bounded everywhere. However, for mappings between sufficiently regular spaces, for instance, spaces of locally bounded geometry, the two definitions coincide. It seems for us more reasonable to require everywhere finiteness and essentially boundedness, rather than everywhere boundedness, for quasiconformal mappings between general metric measure spaces. The importance of quasiconformal mappings in complex analysis was realized by Ahlfors, Teichmuller and Morrey in the 1930s. Ahlfors used quasiconformal mapping ¨ s in his geometric approach to Nevanlinna's value distribution theory that earned him one of the first two Fields medals. Teichmuller used quasiconformal mappings to measure a ¨ distance between two conformally inequivalent compact Riemann surfaces, starting what is now called Teichmuller theory. Morrey proved a powerful existence theorem, called ¨ the measurable Riemann mapping theorem, which has had tremendous impact on complex analysis and dynamics, Teichmuller theory, and low dimensional topology, inverse ¨ problems and partial differential equations. The higher-dimensional theory of quasiconformal mappings was initiated in earnest by Reshetnyak, Gehring and V¨ais¨al¨a in the early 1960s [\[25,](#page-88-1) [26,](#page-89-0) [76\]](#page-90-0). The generalisations to non-injective quasiregular mappings was initiated with Reshetnyak, and the basic theory was comprehensively laid and significantly advanced in a sequence of papers from the Finnish school of Martio, Rickman and V¨ais¨al¨a in the late 1960s [\[62,](#page-90-1) [63,](#page-90-2) [64\]](#page-90-3). Historically, there are three different definitions of quasiconformality, namely, the metric definition, the analytic definition, and the geometric definition (see Section [6.1](#page-39-1) below for the detailed description). Apparently, the metric definition is of infinitesimal flavor, the analytic definition is a point-wise condition, and the geometric definition is more of global nature. It is a rather deep fact, due to Gehring and V¨ais¨al¨a, that all the three definitions of quasiregularity are quantitatively equivalent. The interplay of all three aspects of quasiconformality/quasiregularity is an important feature of the theory; see [\[7,](#page-88-2) [48,](#page-89-1) [76,](#page-90-0) [80,](#page-91-0) [100\]](#page-91-1) for more information on the Euclidean theory of these mappings. In the Euclidean spaces, another remarkable result of Heinonen and Koskela [\[39\]](#page-89-2) implies that in the definition of metrically quasiconformal mappings, the linear dilation H^f (x) can be replaced with h^f (x) = lim infr→⁰ H^f (x, r), which we will refer as weak metrically quasiconformal mappings (see also Section [6.1](#page-39-1) below for the precise formulation). This result was later generalized to mappings between metric spaces with locally bounded geometry in [\[4\]](#page-88-3). The study of quasiconformal mappings beyond Riemannian spaces was first appeared in the celebrated work of Mostow [\[69\]](#page-90-4) on strong rigidity of locally symmetric spaces. The boundary of rank-one symmetric spaces can be identified as certain Carnot groups of step two, and Mostow has developed the basic metrically quasiconformal theory in these groups. Inspired by Mostow's work, Pansu [\[73\]](#page-90-5) used the theory of quasiconformal mappings to study quasi-isometries of rank-one symmetric spaces. In particular, he has shown that the metrically quasiconformal mappings are absolutely continuous on almost every lines. The systematic study of metrically quasiconformal mappings on the Heisenberg group was later done by Kor´anyi and Reimann [\[53,](#page-90-6) [54\]](#page-90-7). Margulis and Mostow [\[60\]](#page-90-8) studied the absolute continuity of metrically quasiconformal mappings along horizontal lines in the setting of equiregular subRiemannian manifolds and proved that metrically quasiconformal mappings between two equiregular subRiemannian manifolds are P-differentiable¹ almost everywhere. By the break-through work of Heinonen and Koskela [\[40\]](#page-89-3), a fullfledged metrically quasiconformal mappping theory exists in rather general metric measure spaces. In particular, the equivalence of metrically quasiconformal mappings and quasisymmetric mappings was derived under very mild assumptions. This theory has subsequently been applied to new rigidity studies in geometric group theory, geometric topology and geometric paratrization of metric spaces; see for instance [\[12,](#page-88-4) [13,](#page-88-5) [43,](#page-89-4) [38,](#page-89-5) [75\]](#page-90-9) and the references therein. This theory also initiated a new way of looking at weakly differentiable mappings between non-smooth metric measure spaces. Based on this theory, in [\[41\]](#page-89-6), the Sobolev class of Banach space valued mappings was studied and several characterizations of quasiconformal mappings between metric spaces of locally bounded geometry were established. In particular, the equivalences of all the three definitions of quasiconformality were proved in their setting; see also [\[95,](#page-91-2) [96,](#page-91-3) [97\]](#page-91-4). ¹P-differentiability was refered as cc-differentiability in their paper The study of quasiregular mappings beyond Riemannian spaces was initiated by Heinonen and Holopainen [\[37\]](#page-89-7). The importance of the study of quasiregular mappings beyond the Riemannian spaces was recognized later in the fundamental paper of Heinonen and Rickman [\[43\]](#page-89-4) as well as in the sequential remarkable papers of Heinonen and Sullivan [\[45\]](#page-89-8) and Heinonen and Keith [\[38\]](#page-89-5). Initiated by these works, there are some recent advances of the theory of quasiregular mappings in various other settings; see for instance [\[70,](#page-90-10) [29,](#page-89-9) [28,](#page-89-10) [1\]](#page-88-6). The main obstacle in establishing the theory of quasiregular mappings in general metric spaces lies in the branch set B^f , i.e., the set of points in X where f : X → Y fails to be a local homeomorphism. The difficulty is somehow hidden in the Euclidean planar case, as the celebrated Stoilow factorization theorem (cf. [\[2\]](#page-88-7)) asserts that a quasiregular mapping f : Ω → R ² admits a factorization f = ϕ ◦ g, where g : Ω → g(Ω) is quasiconformal and ϕ: g(Ω) → R 2 is analytic. This factorization, together with our more or less complete understanding of the structure of analytic functions in the plane, connects quasiregular and quasiconformal mappings strongly. In particular, the branch set B^f of f must be discrete. In higher dimensions or more general metric measure spaces, the branch set of a quasiregular mappings can be very wild. This makes the homeomorphic theory and the non-homeomorphic theory substantially different. Indeed, the most delicate part of establishing the theory of quasiregular mappings in various settings as mentioned above is to show that the branch set and its image have null measure. For instance, in the very recent paper [\[23\]](#page-88-8), in order to derive the basic properties of quasiregular mappings from that of quasiconformal mappings in equiregular subRiemannian manifolds, it was assumed a priori in the definition of a quasiregular mapping that both the branch set and its image are null sets with respect to the underlying Hausdorff measure. Somewhat surprisingly, in our previous paper [\[30\]](#page-89-11), we have successfully generalized the well-known Bonk–Heinonen–Sarvas Theorem, regarding the quantitative porosity of the branch set and its image of a quasiregular mapping, to a large class of metric spaces. One important point there was that we had an argument to prove the porosity results on B^f and f(B^f ) directly, without knowing a priori the deep analytic facts (such as differentiability, Condition N and Condition N ⁻¹ ) of quasiregular mappings, so that many basic properties (in particular those local properties) follow directly from that of quasiconformal mappings. In other words, we have provided a quick approach to establish the basic properties of quasiregular mappings from that of quasiconformal mappings. The result we have obtained also promotes the important V¨ais¨al¨a's inequality in most general settings. On the other hand, to obtain the preceding null property of the branch set and its image with respect to the given measures, one necessarily has to impose certain extra assumptions from quantitative topology for the underlying metric measure spaces that we do want to dispose of. Indeed, our prime motivation of this paper is to prove the mutual implications of different definitions of quasiregularity in the most general metric space setting. Our starting point is to find a powerful factorization result, similar as the Stoilow factorization theorem in the plane, for quasiregular mappings in general metric measure spaces, so that the theory of quasiconformal mappings can be employed. Fortunately, such kind of factorization do exist and we will call it the pullback factorization (see Section [5](#page-30-0) below). More precisely, given a quasiregular mapping f : X → Y between two locally compact complete metric measure spaces, there exists a canonical factorization f = π ◦ g, where g : X → X^f is a quasiconformal mapping and π : X^f → Y is a mapping of 1-bounded length distortion (1-BLD for short). Due to the fine properties of the projection mapping π, f and g share many common analytic and geometric properties. Indeed, one of the key steps in our main equivalence theorems in Section [6](#page-39-0) is to characterize the geometric KO-inequality and K^I -inequality (also known as the Poletsky's inequality) of f via the corresponding inequalities of g. There are two new observations on the theory of quasiregular mappings, even in Euclidean spaces, out from the pullback factorization. First, there are two natural groups, namely, the forward group and the inverse group, each consists of four different definitions of quasiregularity. To gain some intuition of what is going on here, let us, for simplicity, only look at the quasiconformal case. The forward group consists of the following four definitions: weak metric definition (m), metric definition (M), analytic definition (A), and the KO-inequality (G). While the inverse group consists of the "inverse" definitions: the weak inverse metric definition (m^∗ ), the inverse metric definition (M^∗ ), the inverse analytic definition (A^∗ ), and the K^I -inequality (G^∗ ). It is useful to observe that the inverse group is exactly the forward group for f −1 . From the technical point of view, the group in this way makes the implications clear, namely, (A) ⇐⇒ (G) and (A ∗ ) ⇐⇒ (G^∗ ) in arbitrary metric measure spaces, a stronger version of (m) or (m^∗ ) will quantitatively imply both (A) and (A^∗ ) in Ahlfors regular spaces, and all these eight definitions will be quantitatively equivalent when the metric measure spaces have locally bounded geometry. We will use exactly the same groups in the quasiregular case, but some care need to pay to define the inverse analytic definition (A^∗ ) of f due to the branching. This will be done with the help of the pullback factorization. The novel here is that the equivalence of (A) ∗ and G^∗ provides an analytic characterization of the Poletsky's inequality, which together with KO-inequality and the assumption ν(f(B^f )) = 0 will imply the stronger V¨ais¨al¨a's inequality (see Theorem [6.42\)](#page-65-1). Secondly, in Ahlfors regular spaces, for the implication of (a stronger form of) (m) or (m^∗ ) to (A) and (A^∗ ), we do not really need the fact that the branch set (and its image) are null with respect to the Ahlfors regular measures, but that for the essential branch set, which is a subset of the branch set and consists of all points with local essential index strictly larger than one. If the underlying metric measure spaces are doubling, then one can show that the essential branch set of f is null with respect to certain pullback measure, which is sufficient to conclude our implications in Ahlfors regular spaces. 1.1. Main results. Our first main result concerns the relations between all the different definitions of quasiregularity. It can be regarded as a complete analog of that of quasiconformal mappings [\[41,](#page-89-6) [4\]](#page-88-3). Theorem A. Let f : X → Y be a onto branched covering between two metric measure spaces and Q > 1. Then the following conclusions hold: - i). f is analytically KO-quasiregular with exponent Q if and only if it is geometrically KO-quasiregular with exponent Q. Similarly, f is inverse analytically K^I quasiregular with exponent Q if and only if it is strong inverse geometrically K^I quasiregular with exponent Q. - ii). If X and Y are locally Ahlfors Q-regular, and Y has c-bounded turning, then either of the following two conditions - a). h^f (x) ≤ h for all x ∈ X; - b). h ∗ f (x) ≤ h for all x ∈ X, - implies that f is analytically KO-quasiregular with exponent Q and inverse analytically K^I -quasiregular with exponent Q, with both constants K^O and K^I depending only on the constant of Ahlfors Q-regularity, and on c and h. - iii). If both X and Y have locally Q-bounded geometry, then all of the metric, geometric and analytic definitions are quantitatively equivalent. - iv). If f satisfies the K^I -inequality with exponent Q and if ν(f(B^f )) = 0, then f satisfies the V¨ais¨al¨a's inequality with exponent Q and with the same constant. A fundamental observation that we have made use of, when trying to move from the metric definition to the analytic one, is that the use of balls in the typical covering argument (see e.g. [\[40,](#page-89-3) [4,](#page-88-3) [105\]](#page-92-0)) to contruct an appropriate upper gradient is not essential. Alternatively, one can use certain normal neighborhoods, which are obtained as preimages of balls from the target, to run a similar covering type argument. This observation naturally motivates the study of pullback geometry/topology as we have developed in Section [4,](#page-18-0) which plays a crucial role in the discover of the pullback factorization. Another point we would like to point out here is that based on the pullback factorization, we have introduced the class of branched quasisymmetric mappings, which is branched version of the well-known quasisymmetric mappings. As in the case of quasiconformal mappings, we have shown in Section [6.7](#page-68-0) that quasiregular mappings are locally branched quasisymmetric, quantitatively, in metric spaces of bounded geometry. As the first main application of the pullback factorization, we get our second main result of this paper (see Section [5](#page-30-0) below for the definition of BDD branched coverings). Theorem B. Let f : X → Y be a BDD branched covering, with N = N(f, X) < ∞, and suppose that Y is doubling and that X and Y have bounded turning. Then there is a bi-Lipschitz embedding of X into Y × R ^cd(N−1), where c^d depends only on the data of Y . An immediate corollary of Theorem [B](#page-6-0) (cf. Corollary [8.2\)](#page-75-2) is that every BLD covering of a locally compact quasiconvex metric space over R ⁿ with finite maximal multiplicity embeds bi-Lipschitzly into some Euclidean space. This answers in the affirmative an open question of Heinonen and Rickman [\[43\]](#page-89-4). With the same technique, we also construct in Section [8.3](#page-79-0) some natural examples of BLD mappings (even 1-BLD), between sufficiently nice metric spaces, with positive branch sets (as well as their images). This shows the sharpness of the main results of [\[30\]](#page-89-11) and also disproves a conjecture of Heinonen and Rickman [\[43\]](#page-89-4). As another application of the pullback techniques, we fully characterize BLD mappings, via quasiregular mappings, in metric spaces with (locally) bounded geometry. Theorem C. Let f : X → Y be a branched covering between two Ahlfors Q-regular Q-Loewner spaces wtih Q > 1. Then the following statements are equivalent: - 1). f is L-BLD; - 2). For each x ∈ X, there exists r^x > 0 such that $$\frac{d(x,y)}{c} \le d(f(x), f(y)) \le cd(x,y)$$ for all y ∈ B(x, rx); - 3). L^f (x) ≤ c and l^f (x) ≥ 1 c for each x ∈ X; - 4). f is metrically H-quasiregular, locally M-Lipschitz, and J^f (x) ≥ c for a.e. x ∈ X. Moreover, all the constants involved depend quantitatively only on each other and on the data associated to X and Y . As a consequence of Theorem [C,](#page-7-1) we obtain the important compactness properties of BLD mappings between certain generalized manifolds (see Theorem [9.2\)](#page-83-0). 1.2. Structure of the paper. After the introduction, we collect some basic definitions and notation in Section [2.](#page-9-0) In Section [4,](#page-18-0) we study some basic properties of pullback geometry. The main task is to introduce the essential local index and prove some basic estimates on the size of the essential branch set. Section [5](#page-30-0) is the heart of this paper, from the point of view of techniques. We first present the general construction of the pullback factorization for branched coverings between locally compact complete metric measure spaces. Then we introduce the so-called mappings of bounded diameter distortion (BDD for short) and investigate the nice properties of the projection mapping π and also the pullback metric in details. Some basic properties between our branched covering f and the lifting mapping g are also studied. Section [3](#page-13-0) contains a brief introduction of the theory of Sobolev spaces on metric measure spaces based on the upper gradients. The main body of this paper lies in Section [6,](#page-39-0) where we establish the foundations of the theory of quasiregular mappings in general metric measure spaces and prove our first main result Theorem [A.](#page-5-1) More precisely, we introduce nine different definitions of quasiregularity in Section [6.1](#page-39-1) and then show in Section [6.2](#page-41-0) the equivalences of (A) and (G) (and also (A ∗ ) and (G^∗ )) in arbitrary metric measure spaces. Section [6.3](#page-48-0) is the most technical part of the whole Section [6,](#page-39-0) where we show that when the metric measure spaces are Ahlfors regular with the target space having bounded turning, then the (everywhere finite) weak metric/metric\* definitions of quasiregularity implies both the analytic and the inverse analytic definitions, quantitatively. The key for our argument lies in Theorem [6.21,](#page-49-0) which essentially dates back to Balogh–Koskela–Rogovin [\[4\]](#page-88-3). The principle idea is that the (weak) metric dilatation gives good control of balls with small radius that is sufficient, via a standard covering argument similar as [\[4\]](#page-88-3), to construct appropriate weak upper gradients for the mapping itself in Ahlfors regular spaces. Moreover, one can quantifies the precise bound, which will result in the analytic or inverse analytic definitions of quasiregularity. One technical point here is that we need to run Theorem [6.21](#page-49-0) twice to get rid of the dependences on the local essential index for the quasiregularity constant. Finally, we show the equivalences of the nine definitions of quasiregularity in metric measure spaces with locally bounded geometry in Section [6.4.](#page-55-0) Section [6.5](#page-65-0) contains a brief discussion on the size of the branch set, mainly the results from [\[30\]](#page-89-11), and the V¨ais¨al¨a's inequality. In particular, we show that when f satisfies the Poletsky's inequality, and the image of the branch set has zero measure, then f satisfies the V¨ais¨al¨a's inequality. In Section [6.6,](#page-66-0) we study homeomorphisms with finite (weak) linear dilatation and derive some weighted K^O or K^I inequalities for such mappings. In the final section, Section [6.7,](#page-68-0) we introduce the class of branched quasisymmetric mappings and prove that metrically quasiregular mappings are locally branched quasisymmetric, quantitatively, in spaces of locally bounded geometry. Section [7](#page-71-0) contains a brief overview of the foundations of quasiregular mappings in equiregular subRiemannian manifolds. We separate the V¨ais¨al¨a's inequality as a typical application of our main results from Section [6.](#page-39-0) We prove our most important applications of the pullback techniques, Theorem [B,](#page-6-0) in Section [8.](#page-75-0) We also use our preceding techniques to construct natural examples of BLD mappings between very nice metric spaces with positive branch set. The proof of our second application, Theorem [C,](#page-7-1) is given in Section [9,](#page-82-0) where we also prove a compactness results for BLD mappings in great generality. 1.3. An important clarification from the first-named author. The new tool, namely, the pullback factorization, as well as the main results (in Sections [6](#page-39-0) and Section [8\)](#page-75-0) in this paper were essentially obtained in December 2014 by the second-named author in two short notes [\[106,](#page-92-1) [107\]](#page-92-2). The first-named author essentially only combined these notes together in a more detailed and readable manner, together with our joint paper [\[31\]](#page-89-12) and a few natural applications of these techniques. Thus he wants to point out the credits of this paper should be given to the second-named author. Acknowledgements. Some of this research was conducted by the second-named author at the Institute for Pure and Applied Mathematics during the program "Interactions Between Analysis and Geometry" 2013.

Definition 2.1. A space X is called an n-dimensional, n ≥ 2, cohomology manifold (over Z), or a cohomology n-manifold if - (a): the cohomological dimension dim^Z X is at most n, and - (b): the local cohomology groups of X are equivalent to Z in degree n and to zero in degree n − 1. Condition (a) means that H^p c (U) = 0 for all open U ⊂ X and p ≥ n + 1. Condition (b) means that for each point x ∈ X, and for each open neighborhood U of x, there is another open neighborhood V of x contained in U such that $$H_c^p(V) = \begin{cases} \mathbb{Z} & \text{if } p = n \\ 0 & \text{if } p = n - 1, \end{cases}$$ and the standard homomorphism $$(2.1) H_c^n(W) \to H_c^n(V)$$ is a surjection whenever W is an open neighborhood of x contained in V . As for examples of cohomology n-manifolds, we point out all topological n-manifolds are cohomology nmanifolds. More examples can be found in [\[43\]](#page-89-4).

Definition 2.2. A space X is called a generalized n-manifold, n ≥ 2, if it is a finitedimensional cohomology n-manifold. If a generalized n-manifold X satisfies Hⁿ c (X) ≃ Z, then X is said to be orientable and a choice of a generator g^X in Hⁿ c (X) is called an orientation; X together with g^X is an oriented generalized n-manifold. If X is oriented, we can simultaneously choose an orientation g^U for all connected open subsets U of X via the isomorphisms $$H_c^n(U) \to H_c^n(U)$$ . Let X and Y be oriented generalized n-manifolds, Ω ⊂ X be an oriented domain and let f : Ω → Y be continuous. For each domain D ⊂⊂ Ω and for each component V of Y \f(∂D), the map $$f|_{f^{-1}(V)\cap D}: f^{-1}(V)\cap D\to V$$ is proper. Hence we have a sequence of maps (2.2) $$H_c^n(V) \to H_c^n(f^{-1}(V) \cap D) \to H_c^n(D),$$ where the first map is induced by f and the second map is the standard homomorphism. The composition of these two maps sends the generator g^V to an integer multiple of the generator gD; this integer, denoted by µ(y, f, D), is called the local degree of f at a point y ∈ V with respect to D. The local degree is an integer-valued locally constant function $$y \mapsto \mu(y, f, D)$$ defined in $Y \setminus f(\partial D)$ . If $V \cap f(D) = \emptyset$ , then $\mu(y, f, D) = 0$ for all $y \in V$ .

Definition 2.3. A continuous map $f: X \to Y$ between two oriented generalized n-manifolds is said to be sense-preserving if $$\mu(y, f, D) > 0$$ whenever $D \subset\subset X$ is a domain and $y \in f(D) \setminus f(\partial D)$ . 2.2. Hausdorff measure. Let X = (X, d) be a metric space. Fix a positive real number s. For each $\delta > 0$ and $E \subset X$ , we set $$\mathcal{H}_{s,\delta} = \inf \sum_{i} (\operatorname{diam}(E_i))^s,$$ where the infimum is taken over all countable covers of E by sets $E_i \subset X$ with diameter no more than $\delta$ . When $\delta$ decreases, the value of $\mathcal{H}_{s,\delta}(E)$ for a fixed set E increases, and the s-dimensional Hausdorff measure of E is defined to be (2.3) $$\mathcal{H}^{s}(E) := \lim_{\delta \to 0} \mathcal{H}_{s,\delta}(E).$$ The set function $E \mapsto \mathcal{H}^s(E)$ determins a Borel regular measure on X. In the later part of this paper, we also use the notation $\mathscr{H}^s(E)$ for the s-dimensional Hausdorff measure of a set E in a metric space X. The Hausdorff dimension of a set E in X is the infimum of the numbers s > 0 such that $\mathcal{H}^s(E) = 0$ .

Definition 2.4. A metric measure space is defined to be a triple $(X, d, \mu)$ , where (X, d) is a separable metric space and $\mu$ is a nontrivial locally finite Borel regular measure on X.

Definition 2.5. A Borel regular measure $\mu$ on a metric space (X, d) is called a doubling measure if every ball in X has positive and finite measure and there exists a constant $C_{\mu} \geq 1$ such that (2.4) $$\mu(B(x,2r)) \le C_{\mu}\mu(B(x,r))$$ for each $x \in X$ and r > 0. We call the triple $(X, d, \mu)$ a doubling metric measure space if $\mu$ is a doubling measure on X. We call $(X, d, \mu)$ an Ahlfors Q-regular space, $1 \le Q < \infty$ , if there exists a constant $C \ge 1$ such that (2.5) $$C^{-1}r^{Q} \le \mu(B(x,r)) \le Cr^{Q}$$ for all balls $B(x,r) \subset X$ of radius $r < \operatorname{diam} X$ . It is well-known that if $(X, d, \mu)$ is an Ahlfors Q-regular space, then (2.6) $$\mu(E) \approx \mathcal{H}^Q(E)$$ for all Borel sets E in X, see [34, Chapter 8]. 2.4. Local contractibility and local metric orientation. Recall that a metric space X is said to be linearly locally contractible if there exits a constant $c \ge 1$ such that every ball B(x,r) is contractible in B(x,cr). X is locally linearly locally contractible if for each $x \in X$ , there exists a neighborhood $U_x$ of x such that $U_x$ is $c_x$ -linearly locally contractible. Assume that X is an oriented generalized n-manifold, and assume that X satisfy the following conditions: - X is n-rectifiable and has locally finite $\mathcal{H}^n$ -measure; - X is Ahlfors n-regular; - X is locally bi-Lipschitz embeddable in Euclidean space. Let U be an open connected neighborhood of a point in X that can be embedded bi-Lipschitz in some $\mathbb{R}^N$ and that U has finite Hausdorff n-measure. Because of properties (A1) and (A2), the set U has a tangent n-plane $T_xU$ at a.e. point $x \in U$ . The collection of these planes is called a measurable tangent bundle of U, and it is denoted by TU. Each n-plane $T_xU$ , whenever it exists, is an n-dimensional subspace of $\mathbb{R}^N$ , and a measurable choice of orientations $\xi = (\xi_x)$ on each $T_xU$ is called an orientation of the tangent bundle TU. Such orientations always exist. Because X is an oriented generalized n-manifold, there is another orientation on U, provided U is connected; this is a generator $g_U$ in the group $H_c^n(U) = \mathbb{Z}$ determined by the fixed orientation on X. Fix a point $x \in U$ such that $T_xU$ exists. Then the projection $$\pi_x: \mathbb{R}^N \to T_x U + x$$ satisfies $x \notin \pi_x(\partial D)$ whenever D is a sufficiently small open connected neighborhood of x in U. Moreover, for any $\varepsilon > 0$ , (2.7) $$\frac{|x - x_0|}{2} \le \pi_{x_0}(x - x_0) \le \varepsilon |x - x_0|$$ whenever $|x - x_0|$ small enough, see e.g. [43, 27]. Thus, if V is the x-component of $(T_xU + x) \setminus \pi_x(\partial D)$ , we have (2.8) $$H_c^n(T_xU) \leftarrow H_c^n(V) \xrightarrow{\pi_x^*} H_c^n(\pi_x^{-1}(V) \cap D) \to H_c^n(D) \to H_c^n(U),$$ where the unnamed arrows represent a canonical isomorphism. If the orientations $\xi_U$ and $g_U$ correspond to each other under the map in (2.8), we say that $T_xU$ and U are coherently oriented at x by $\xi_x$ and $g_U$ ; if a measurable coherent orientation $\xi = (\xi_x)$ is chosen at a.e. point, we say that TU is metrically oriented by $\xi$ and $g_U$ . 2.5. Inverse dilatation. Let $f: X \to Y$ be continuous. For each $x \in X$ , denote by U(x,r) the component of x in $f^{-1}(B(f(x),r))$ . Set $$H_f^(x,s) = \frac{L_f^(x,s)}{l_f^*(x,s)},$$ where $$L_f^*(x,s) = \sup_{z \in \partial U(x,s)} d(x,z)$$ and $l_f^*(x,s) = \inf_{z \in \partial U(x,s)} d(x,z)$ . The inverse linear dilatation function of f at x is defined pointwise by $$H_f^(x) = \limsup_{s \to 0} H_f^(x, s).$$ Similarly, the weak inverse linear dilatation function of f at x is defined pointwise by $$h_f^(x) = \liminf_{s \to 0} H_f^(x, s).$$ 2.6. Condition N and N ⁻¹ . A mapping f : (X, µ) → (Y, ν) between two measure spaces is said to satisfy Condition N if ν(f(E)) = 0 whenever E is a subset of X with µ(E) = 0. Similarly, f satisfies Condition N ⁻¹ if ν(f(E)) > 0 whenever E is a subset of X with µ(E) > 0.

Definition 3.1. A Borel function $g: X \to [0, \infty]$ is called an upper gradient for a map $f: X \to Z$ if for every rectifiable curve $\gamma: [a, b] \to X$ , we have the inequality (3.3) $$\int_{\gamma} g \, ds \ge d_Z(f(\gamma(b)), f(\gamma(a))).$$ If inequality (3.3) merely holds for p-almost every compact curve, then g is called a p-weak upper gradient for f. When the exponent p is clear, we omit it. The concept of upper gradient were introduced in [40]. It was initially called "very weak gradient", but the befitting term "upper gradient" was soon suggested. Functions with p-integrable p-weak upper gradients were subsequently studied in [55], while the theory of Sobolev spaces based on upper gradient was systematically developed in [88] and [42]. By [42, Lemma 5.2.3], f has a p-weak upper gradient in $L^p_{loc}(X)$ if and only if it has an actual upper gradient in $L^p_{loc}(X)$ . A p-weak upper gradient g of f is minimal if for every p-weak upper gradient $\tilde{g}$ of f, $\tilde{g} \geq g$ $\mu$ -almost everywhere. If f has an upper gradient in $L^p_{loc}(X)$ , then f has a unique (up to sets of $\mu$ -measure zero) minimal p-weak upper gradient by the following result from [42, Theorem 5.3.23]. In this situation, we denote the minimal upper gradient by $g_f$ .

Definition 3.3. We say that a metric measure space $(X, d, \mu)$ admits a (1,p)-Poincaré inequality if there exist constants $C \ge 1$ and $\tau \ge 1$ such that (3.5) $$f_B |u - u_B| d\mu \le C \operatorname{diam}(B) \left( f_{\tau B} g^p d\mu \right)^{1/p}$$ for all open balls B in X, for every function $u: X \to \mathbb{R}$ that is integrable on balls and for every upper gradient q of u in X. The (1, p)-Poincaré inequality can be thought of as a requirement that a space contains "many" curves, in terms of the p-modulus of curves in the space. For a complete doubling metric measure space $X = (X, d, \mu)$ supporting a (1, p)-Poincaré inequality, there are many important consequences [42, Section 9]. For example, one has the important Sobolev embedding results as in the Euclidean spaces. We point out the following geometric implications of Poincaré inequalities from [42, Theorem 8.3.2].

Definition 3.6. Let (X, d, µ) be a metric measure space. We say that X has Q-Loewner property if there is a function φ: (0, ∞) → (0, ∞) such that $$\operatorname{Mod}_Q(\Gamma(E, F, X)) \ge \phi(\zeta(E, F))$$ for every non-degenerate compact connected sets E, F ⊂ X, where $$\zeta(E, F) = \frac{\operatorname{dist}(E, F)}{\min{\{\operatorname{diam} E, \operatorname{diam} F\}}}.$$ By [\[40,](#page-89-3) Theorem 3.6], if X is Ahlfors Q-regular, and Q-Loewner, then (3.7) $$\operatorname{Mod}_{Q}(\Gamma(E, F, X)) \ge C \left(\log \zeta(E, F)\right)^{1-Q}$$ when ζ(E, F) is large enough with C depends only on the data of X. By [\[40,](#page-89-3) Corollary 5.13], a complete (or equivalently proper) Ahlfors Q-regular metric measure space that supports a (1, Q)-Poincar´e inequality is Q-Loewner. Following [\[41\]](#page-89-6), we introduce the notion of metric spaces of locally bounded geometry.

Definition 3.7. A metric measure space (X, d, µ) is said to be of locally Q-bounded geometry, Q ≥ 1, if X is separable, pathwise connected, locally compact, and if there exist constants C⁰ ≥ 1, 0 < λ ≤ 1, and a decreasing function ψ : (0, ∞) → (0, ∞) such that each point x ∈ X has a neighborhood U (with compact closure in X) so that - µ(BR) ≤ C0R^Q whenever B^R ⊂ U is a ball of radius R > 0; - ModQ(Γ(E, F, BR)) ≥ ψ(t) whenever B^R ⊂ U is a ball of radius R > 0 and E and F are two disjoint, non-degenerate compact connected sets in BλR with $$\operatorname{dist}(E, F) \le t \cdot \min\{\operatorname{diam} E, \operatorname{diam} F\}.$$ In other words, a pathwise connected, locally compact space is of locally Q-bounded geometry if and only if it is locally uniformly Ahlfors Q-regular and locally uniformly Q-Loewner. In terms of Poincar´e inequality, a pathwise connected, locally compact space is of locally Q-bounded geometry if and only if it is locally uniformly Ahlfors Q-regular and supports a local uniform (1, Q)-Poincar´e inequality. Here by saying locally uniformly Ahlfors Q-regular we mean that there exists a constant C⁰ > 0 such that for each x ∈ M, there is a radius r^x > 0 so that [\(2.5\)](#page-10-4) holds for all 0 < r < r^x with the constant C0, and by saying supporting a local uniform (1, Q)-Poincar´e inequality, we mean that there exists a constant C > 0 such that for each x, there exists a ball B centered at x (with radius depending on x) such that the Poincar´e inequality [\(3.5\)](#page-15-2) with exponent p = Q holds with the constant C. As a particular case, let us point out that every Riemannian n-manifold is of locally n-bounded geometry. More exotic examples can be found in [\[40,](#page-89-3) Section 6].

Definition 4.6. (Admissible pointed neighborhoods). A sequence $\mathcal{A} = \{A_i : x_i \in A_i\}_{i \in I}$ of pointed neighborhoods is admissible if for each $i \neq j$ form I, the following two conditions are satisfied: - (1) Either $x_i \notin A_i$ or $x_i \notin A_i$ . - (2) $A_i \not\subseteq A_i \not\subseteq A_i$ . We have the following simple fact for admissible pointed sets, which slightly generalizes [4, Lemma 2.3].

Definition 5.2. A branched covering $f: X \to Y$ between two metric spaces is said to be an L-BLD, or a mapping of L-bounded length distortion, $L \ge 1$ , if $$L^{-1}l(\alpha) < l(f \circ \alpha) < Ll(\alpha)$$ for all non-constant curves $\alpha$ in X, where $l(\gamma)$ denotes the length of a curve $\gamma$ in a metric space. The definition of BLD mappings is clearly only interesting if the metric spaces X and Y have a reasonable supply of rectifiable curves and so the most natural setting in which we study such mappings is that of quasiconvex metric spaces; we recall that a metric space X is c-quasiconvex if every pair of points $x_1, x_2 \in X$ may be joined with a curve $\gamma$ of length $l(\gamma) \leq cd(x_1, x_2)$ . When the constant c is unimportant, we omit it. Since our construction of pullback metric generally only requires the control of diameters of sets, a more appropriate setting for our results will be the more general class of metric spaces of bounded turning. Recall that X has c-bounded turning if every pair of points $x_1, x_2 \in X$ can be joined by a continuum $E \subset X$ such that diam $E \leq cd(x_1, x_2)$ . Note that by the local connectivity, we also have local path connectivity, and we may use curves instead of general continuua in the definition of bounded turning. It is elementary to verify, as in the case of quasiconvex spaces, that the infimum of the diameters of continuua or curves joining two points is realized, provided X is assumed to be complete as well as locally compact. When X and Y have bounded turning, the natural branched analog of a bi-Lipschitz homeomorphism is what we call a mapping of bounded diameter distortion, which is defined in analogy with BLD mappings:

Definition 5.3. A branched covering $f: X \to Y$ between two metric spaces is said to be an L-BDD, or a mapping of L-bounded diameter distortion, $L \ge 1$ , if $$L^{-1}\operatorname{diam}(\alpha) \leq \operatorname{diam}(f \circ \alpha) \leq L\operatorname{diam}(\alpha)$$ for all non-constant curves $\alpha$ in X. It follows directly from the definition of arc-length that an L-BDD mapping is L-BLD as well, regardless of any connectivity assumptions on either X or Y. We will see later that if Y is c-quasiconvex and f is L-Lipschitz and L-BLD with $N = N(f, X) < \infty$ , then f is LcN-BDD. In particular, every L-BLD mapping $f: X \to Y$ between length spaces is LN-BDD. As we will see in a moment, the metric space $X^f$ retains the original topology of X and is often rather well-behaved: having 1-bounded turning and inheriting many metric and geometric properties from Y. Moreover, the branched covering $\pi$ from the pullback factorization is easily seen to be 1-Lipschitz and 1-BDD (and a fortiori 1-BLD). 5.3. Fine properties of the pullback metric. In this section, we fix a branched covering $\pi: X \to Y$ between two metric spaces. However, we will typically consider the space $X^{\pi}$ (or $\pi^Y$ ) by endowing the set X with the pullback metric $\pi^d_Y$ . Recall also that for each metric d on a topological space X, the length metric $l_d(z_1, z_2)$ is given by infimizing the lengths of all curves joining $z_1$ and $z_2$ . For a metric space $X = (X, d_X)$ , we denote by $X^l$ the length space $(X, l_{d_X})$ . We caution the reader that there are subtleties involved depending on the order in which one pulls back the metric, restricts the mapping to a neighborhood, or passes to the length metric. That is, in general, $l_{\pi^d_Y}$ need not coincide with $\pi^l_{d_Y}$ , nor must $(\pi|_U)^d_Y$ necessarily coincide with $(\pi^d_Y)|_U$ . Nevertheless, we will see in a moment that these distinctions are rather minor. We need the following well-known path-lifting result of Floyd [24, Theorem 2].

Definition 6.1. (Weak metrically quasiregular mappings). A branched covering $f: \tilde{\Omega} \to \Omega$ is said to be weakly metrically H-quasiregular if it satisfies i). $$h_f(x) < \infty$$ for all $x \in \tilde{\Omega}$ ; ii). $h_f(x) \leq H$ for $\mu$ -almost every $x \in \tilde{\Omega}$ . The second definition is very commonly used in literature as it is defined pointwisely and is easy to deduce analytic properties of quasiregular mappings.

Definition 6.2. (Analytically quasiregular mappings). A branched covering $f : \tilde{\Omega} \to \Omega$ is said to be analytically K-quasiregular with exponent Q if $f \in N^{1,Q}_{loc}(\tilde{\Omega},\Omega)$ and $$|\nabla f|(x)^Q \le KJ_f(x)$$ for $\mu$ -a.e. $x \in \tilde{\Omega}$ . The geometric definition requires some modulus inequalities between curve families.

Definition 6.3. (Geometrically quasiregular mappings). A branch covering $f: \tilde{\Omega} \to \Omega$ is said to be geometrically K-quasiregular with exponent Q if it satisfies the $K_O$ -inequality with exponent Q, i.e., for each open set $\tilde{\Omega}_0 \subset \tilde{\Omega}$ and each path family $\Gamma$ in $\tilde{\Omega}_0 \subset \tilde{\Omega}$ , if $\rho$ is a test function for $f(\Gamma)$ , then $$\operatorname{Mod}_{Q}(\Gamma) \leq K \int_{\Omega} N(y, f, \tilde{\Omega}_{0}) \rho^{Q}(y) d\nu(y).$$ As pointed out in the introduction, we will refer to the metric definition (M), the weak metric definition (m), the analytic definition (A), and the geometric definition (G) as elements of the forward definitions. Next, we introduce the elements from the inverse definitions: the inverse metric definition $(M^)$ , the inverse weak metric definition $(m^)$ , the inverse analytic definition $(A^)$ , and the inverse geometric definition $(G^)$ .

Definition 6.4. (Inverse metrically quasiregular mappings). A branched covering $f: \tilde{\Omega} \to \Omega$ between two metric measure spaces is termed inverse metrically H-quasiregular if the inverse linear dilatation function $H_f^*$ is finite everywhere and essentially bounded from above by H.

Definition 6.5. (Inverse weak metrically quasiregular mappings). A branched covering $f \colon \tilde{\Omega} \to \Omega$ between two metric measure spaces is termed inverse weak metrically H-quasiregular if it satisfies - i). $h_f^*(x) < \infty$ for all $x \in \tilde{\Omega}$ ; - ii). $h_f^*(x) \leq H$ for $\mu$ -almost every $x \in \tilde{\Omega}$ . We need the pullback factorization to define the inverse analytically quasiregular mappings and the name of this terminology will become clear soon.

Definition 6.6. (Inverse analytically quasiregular mappings). A branched covering $f : \tilde{\Omega} \to \Omega$ is said to be inverse analytically K-quasiregular with exponent Q if $g^{-1} \in N^{1,Q}_{loc}(\tilde{\Omega}^f, \tilde{\Omega})$ and $$|\nabla q^{-1}|(z)^Q < KJ_{q^{-1}}(z)$$ for $\lambda$ -a.e. $z \in \tilde{\Omega}^f$ . The inverse geometric definition also relies on certain inequalities for the modulus of curve families.

Definition 6.7. (Inverse geometric quasiregular mappings). A branched covering $f: \tilde{\Omega} \to \Omega$ is said to be inverse geometrically K-quasiregular with exponent Q if it satisfies the $K_I$ -inequality or the Poletsky's inequality with exponent Q, i.e., for every curve family $\Gamma$ in $\tilde{\Omega}$ , we have $$\operatorname{Mod}_Q(f(\Gamma)) \leq K \operatorname{Mod}_Q(\Gamma).$$ We also introduce the following strong inverse geometrically quasiregular mappings and it will be useful in our later proofs of the standard Väisälä's inequality.

Definition 6.8. (Strong inverse geometrically quasiregular mappings). A branched covering $f: \tilde{\Omega} \to \Omega$ is said to be a strong inverse geometric K-quasiregular mapping with exponent Q if it satisfies the following generalized Väisälä's inequality with exponent Q: For each open subset $\tilde{\Omega}_0 \subset \tilde{\Omega}$ , each curve family $\Gamma$ in $\tilde{\Omega}_0$ , $\Gamma'$ in $\Omega$ , and for each $\gamma' \in \Gamma'$ , there are curves $\gamma_1, \ldots, \gamma_m \in \Gamma$ and subcurves $\gamma'_1, \ldots, \gamma'_m$ of $\gamma'$ such that for each $i = 1, \ldots, m$ , $\gamma'_i = f(\gamma_i)$ , and for almost every $s \in [0, l(\gamma')], \gamma_i(s) = \gamma_j(s)$ if and only if i = j, then $$\operatorname{Mod}_Q(\Gamma') \leq \frac{K}{m} \operatorname{Mod}_Q(\Gamma).$$ By [41, Theorem 9.8], when $f: \tilde{\Omega} \to \Omega$ is a homeomorphism, and $\tilde{\Omega}$ and $\Omega$ have locally Q-bounded geometry, the inverse (metric, weak metric, analytic, geometric) definitions for f are, quantitatively, the forward (metric, weak metric, analytic geometric) definitions for $f^{-1}$ . Moreover, in this case, each of these definitions is further equivalent to the local quasisymmetry, quantitatively. 6.2. Analytic and geometric definitions. When $f: \tilde{\Omega} \to \Omega$ is a homeomorphism, the equivalence of the analytic and geometric definitions has been shown in great generality (cf. [104, Theorem 1.1]), with the same dilatation $K_O$ . With our decomposition, we can expand this almost immediately to the general case; in fact, it turns out that f satisfies the $K_O$ inequality with exponent Q if and only if its lift g does. For the "inverse" geometric definition, the situation is a bit more involved, but likewise the inequality holds for f if and only if it holds for g. Our main result of this section is the following very general equivalence result.

Definition 6.29. (Good branching property). The projection $\pi \colon \tilde{\Omega}^f \to \Omega$ is said to have the good branching property if the following holds: Let $\Gamma$ and $\Gamma'$ be families of curves in $\tilde{\Omega}^f$ and $\Omega$ , respectively. Suppose for each $\gamma' \in \Gamma'$ , there are curves $\gamma_1, \ldots, \gamma_m \in \Gamma$ and subcurves $\gamma'_1, \ldots, \gamma'_m$ of $\gamma'$ such that for each $i = 1, \ldots, m, \ \gamma'_i = \pi(\gamma_i)$ , and for a.e. $s \in [0, l(\gamma')], \ \gamma_i(s) = \gamma_j(s)$ if and only if i = j. For each $y \in \Omega$ and 0 < r < s, we let $\mathcal{A}(y, r, s)$ denote the family of curves in B(y, s) joining $\partial B(y, r)$ and $\partial B(y, s)$ . We need the following rather standard modulus estimates.

Definition 6.44. (Quasisymmetric mappings). Let $\eta: [0, \infty) \to [0, \infty)$ be a homeomorphism. A homeomorphism $f: X \to Y$ is $\eta$ -quasisymmetric if (6.21) $$\frac{d(f(x), f(y))}{d(f(x), f(z))} \le \eta \left(\frac{d(x, y)}{d(x, z)}\right)$$ for all distinct triple $x, y, z \in X$ . The class of $\eta$ -quasisymmetric mappings was introduced by Tukia and Väisälä [94] in their study of geometric embeddings of metric spaces. The importance of these mappings was soon realized in the study of (metrically) quasiconformal mappings; we refer the interested readers to the fundamental paper of Heinonen and Koskela [40] for more information on these development. We next define a proper subclass of metrically quasiregular mappings that carry similar global metric information, but less restrictive as those BDD mappings.

Definition 6.45. (Branched quasisymmmetric mappings). Let $f: X \to Y$ be a branched covering. We say that f is branched quasisymmetric (BQS) if there exists a homeomorphism $\eta: [0, \infty) \to [0, \infty)$ such that (6.22) $$\frac{\operatorname{diam} f(E)}{\operatorname{diam} f(F)} \le \eta \left(\frac{\operatorname{diam} E}{\operatorname{diam} F}\right)$$ for all intersected continua $E, F \subset X$ .

Definition 6.46. (Generalized quasisymmetric mappings). A homeomorphism $f: X \to Y$ is generalized $\eta$ -quasisymmetric if it is branched $\eta$ -quasisymmetric. We have the following concrete characterization of branched quasisymmetric mappings via the pullback factorization.

Definition 7.1. (subRiemannian manifold). An equiregular subRiemannian manifold is a triple $(M, \mathcal{D}, g)$ where M is a smooth and connected manifold, $\mathcal{D} \subset TM$ is a bracket generating equiregular subbundle, and g is a smooth inner product on the fibers $\mathcal{D}_p$ , $p \in M$ , of $\mathcal{D}$ . The inner product g is called a horizontal metric of $\mathcal{D}$ . We use the notation $|v|_{g_p}$ or $||v||_{g_p}$ for the norm $\sqrt{g_p(v,v)}$ of a horizontal vector $v \in \mathcal{D}_p$ . When there is no risk of confusion, we sometimes remove the subscript and write simply |v| or $|v|_q$ etc.

Definition 7.2. (subRiemannian distance). An absolutely continuous curve $\gamma \colon [0,1] \to M$ is called a horizontal curve if $\gamma'(t) \in \mathcal{D}_{\gamma(t)}$ for almost every $t \in [0,1]$ . The length $l(\gamma)$ of a horizontal curve $\gamma \colon [0,1] \to M$ is $$l(\gamma) := \int_0^1 \|\gamma'(t)\| \, \mathrm{d}t.$$ The subRiemannian distance is defined by: $$d_g(p,q) := \inf_{\gamma} \{l(\gamma) : \gamma \text{ is a horizontal curve joining } p \in M \text{ to } q \in M\}.$$ An equiregular subRiemannian manifold M can be endowed in a canonical way with a smooth volume $Vol_M$ that is called Popp measure. The construction can be found in [5, 67]. Moreover, when edowed with the subRiemannian distance and the Popp volume measure, an equiregular subRiemannian manifold $M = (M, d, Vol_M)$ becomes a geodesic metric measure space. For each $x_0 \in M$ , there exists a neighborhood U of $x_0$ such that $(U, d, \operatorname{Vol}_M)$ has locally Q-bounded geometry. But we caution that it is not known whether the data associated to the local Q-bounded geometry at each point is uniformly bounded. Namely, for two dist points x and y, both the metric spaces $(U, d, \operatorname{Vol})$ and $(V, d, \operatorname{Vol})$ has locally Q-bounded geometry, but the associated data (for locally Q-bounded geometry) might depend on U and V, respectively. All the different definitions of quasiregularity, as introduced in Section 6.1, directly make senses in the subRiemannian manifolds. However, there is a "better" definition of quasiregularity that reflects the geometry of subRiemannian manifolds. This definition was introduced in [28]³.

Definition 7.3. (Horizontally K-quasiregular mappings) Let $f:(M,g) \to (N,h)$ be a branched covering between equiregular subRiemannian manifolds (M,g) and (N,h). We say that the mapping f is horizontally K-quasiregular with exponent Q if $f \in N^{1,Q}_{loc}(M,N)$ and it satisfies $$||g^{-1}f^h||^r \le K \det(g^{-1}f^h)$$ a.e. in $M$ . The norm $||g^{-1}f^h||$ is the sup-norm of $g^{-1}f^h$ . We remark that the horizontal $\binom{1}{1}$ -tensor $g^{-1}f^*h$ is the same as $$Df^*Df:(\mathcal{D}_M,q)\to(\mathcal{D}_N,h),$$ where the adjoint $Df$ is defined as the adjoint of Df between inner product spaces $(\mathcal{D}_M, g)$ and $(\mathcal{D}_N, h)$ , cf. [57]. We also note that we can regard $g^{-1}f^h$ as an element of $\operatorname{End}(\mathcal{D}_M)$ implying that the eigenvalue problem for $g^{-1}f^*h$ is well defined, i.e. independent of the choice of basis for $\mathcal{D}_M$ . The following result was obtained very recently by Liimatainen and the first-named author [28]. ^&lt;sup>3</sup>There is one more equivalent definition, the so-called subRiemannian quasiregular mappings, introduced in [28], but the formulation requires the local Popp extension and so we do not include it here.