Bers Slices in Families of Univalent Maps

Proposition 2.14. Let $\Gamma$ be a necklace group. Then $\mathcal{F}_{\Gamma}$ is a fundamental domain for $\Gamma$ . Proof. Let $\mathcal{P}_{\Gamma}$ be the convex hyperbolic polyhedron (in $\mathbb{H}^3$ ) whose relative boundary in $\mathbb{H}^3$ is the union of the hyperplanes $S_i$ (see Remark 2.9). Then, by [VS93, Part II, Chapter 5, Theorem 1.2], $\mathcal{P}_{\Gamma}$ is a fundamental domain for the action of $\Gamma$ on $\mathbb{H}^3$ . It now follows that $\mathcal{F}_{\Gamma} = \overline{\mathcal{P}_{\Gamma}} \cap \Omega(\Gamma)$ (where the closure is taken in $\Omega(\Gamma) \cup \mathbb{H}^3$ ) is a fundamental domain for the action of $\Gamma$ on $\Omega(\Gamma)$ [Mar07, §3.5]. It will be useful in our discussion to have a canonical interior necklace group to refer to:

Lemma 2.22. Let $\mu \in \operatorname{Bel}_{\Gamma_d}$ , and $\tau_{\mu} : \mathbb{C} \to \mathbb{C}$ an integrating map. Then $\tau_{\mu} \circ \Gamma_d \circ \tau_{\mu}^{-1}$ is a necklace group. Proof. By definition, the maps $\tau_{\mu} \circ \rho_{i} \circ \tau_{\mu}^{-1}$ generate the group $\tau_{\mu} \circ \mathbf{\Gamma}_{d} \circ \tau_{\mu}^{-1}$ . By invariance of $\mu$ , each $\tau_{\mu} \circ \rho_{i} \circ \tau_{\mu}^{-1}$ is an anti-conformal involution of $\widehat{\mathbb{C}}$ , hence an anti-Möbius transformation. Since $\tau_{\mu} \circ \rho_{i} \circ \tau_{\mu}^{-1}$ fixes $\tau_{\mu}(C_{i})$ , and interchanges its two complementary components, it follows that $\tau_{\mu}(C_{i})$ is a Euclidean circle and hence $\tau_{\mu} \circ \rho_{i} \circ \tau_{\mu}^{-1}$ is reflection in the circle $\tau_{\mu}(C_{i})$ . One readily verifies that the circles $\tau_{\mu}(C_{i})$ satisfy the conditions of Definition 2.11, and so the result follows.

Proposition 2.23. The Bers slice $\beta(\Gamma_d)$ is pre-compact in $\mathcal{D}(\Gamma_d)$ , and for each $\xi \in \overline{\beta(\Gamma_d)}$ , the group $\xi(\Gamma_d)$ is a necklace group. Proof. Let $(\xi_n)_{n=1}^{\infty}$ be a sequence in $\beta(\mathbf{\Gamma}_d)$ , and $\tau_n : \mathbb{C} \to \mathbb{C}$ the associated quasiconformal maps as in Definition 2.20. Since each $\tau_n$ is conformal in $\mathbb{D}$ and is hydrodynamically normalized, by a standard normal family result (see [CG93, Theorem 1.10] for instance) there exists a conformal map $\tau_{\infty}$ of $\mathbb{D}$ such that $\tau_n \to \tau_{\infty}$ uniformly on compact subsets of $\mathbb{D}^*$ , perhaps after passing to a subsequence which we reenumerate $(\tau_n)$ . By Lemma 2.22, each $\tau_n(\mathbf{C}_i)$ is a Euclidean circle which we denote by $C_i^n$ . Since $\tau_n \to \tau_\infty$ uniformly on compact subsets of $D$ , $\tau_\infty(C_i \cap \mathbb{D}^)$ must be a subarc of a Euclidean circle which we denote by $C_i^\infty$ . Denote furthermore by $\rho_i^n$ , $\rho_i^\infty$ the reflections in the circles $C_i^n$ , $C_i^\infty$ (respectively), and by $\Gamma_\infty$ the group generated by reflections in the circles $(C_i^\infty)_{i=1}^d$ . Let $\xi_\infty : \mathbf{\Gamma}_d \to \Gamma_\infty$ be the homomorphism defined by $\xi_\infty(\rho_i) := \rho_i^\infty$ . We see that $C_i^n \to C_i^\infty$ as $n \to \infty$ in the Hausdorff sense, whence it follows that $\rho_i^n \to \rho_i^\infty$ . This proves algebraic convergence $(\xi_n) \to \xi_\infty$ . Hausdorff convergence of $C_i^n \to C_i^\infty$ also implies that each $C_i^\infty$ intersects tangentially with $C_{i+1}^\infty$ , so that $\xi_\infty$ is weakly type preserving. Similar considerations show that the circles $C_i^\infty$ have pairwise disjoint interiors, and the boundary of the unbounded component of $\widehat{\mathbb{C}} \setminus \bigcup_i C_i^\infty$ intersects each $C_i^\infty$ . In particular, there cannot be any non-tangential intersection among the circles $C_i^\infty$ . It now follows that $\xi_\infty$ is indeed an isomorphism, and $\Gamma_\infty = \xi_\infty(\mathbf{\Gamma}_d)$ is a necklace group.

Proposition 2.28. Let $\Gamma \in \overline{\beta(\Gamma_d)}$ . Then the following hold true. - (1) $\Omega_{\infty}(\Gamma)$ is simply connected, and $\Gamma$ -invariant. - (2) $\partial \Omega_{\infty}(\Gamma) = \Lambda(\Gamma)$ . - (3) $\Lambda(\Gamma)$ is connected, and locally connected. - (4) All bounded components of $\Omega(\Gamma)$ are Jordan domains.

Proposition 2.31. Let $\Gamma \in \overline{\beta(\Gamma_d)}$ . The map $\rho_{\Gamma}$ is orbit equivalent to $\Gamma$ on $\widehat{\mathbb{C}}$ . Proof. Suppose $z, w \in \widehat{\mathbb{C}}$ are such that there exist $n_1, n_2 \in \mathbb{N}$ such that $\rho^{n_1}(z) = \rho^{n_2}(w)$ . Since $\rho_{\Gamma}$ acts by the generators $r_i$ of the group $\Gamma$ , it follows directly that there exists $g \in \Gamma$ with g(z) = w. Conversely, let $z, w \in \widehat{\mathbb{C}}$ be such that there exists $g \in \Gamma$ with g(z) = w. By definition, we have that $g = r_{s_1} r_{s_2} \cdots r_{s_n}$ , for some $s_1, \dots s_n \in \{1, \dots, d\}$ . Suppose first that n = 1. Note that either z or w must belong to $\overline{\operatorname{int} C_{s_1}}$ . Since $r_{s_1}(z) = w$ implies $r_{s_1}(w) = z$ , there is no loss of generality in assuming that $z \in \overline{\operatorname{int} C_{s_1}}$ . Now, the condition $r_{s_1}(z) = w$ can be written as $\rho_{\Gamma}(z) = w$ . The case n > 1 now follows by induction. Notation 2.32. For $\Gamma \in \overline{\Gamma_d}$ , we will denote by $T^o(\Gamma)$ the union of all bounded components of the fundamental domain $\mathcal{F}_{\Gamma}$ (see Proposition 2.14), and by $\Pi^o(\Gamma)$ the unique unbounded component of $\mathcal{F}_{\Gamma}$ . We also set $$T(\Gamma) := \overline{T^o(\Gamma)}$$ , and $\Pi(\Gamma) := \overline{\Pi^o(\Gamma)}$ . Remark 2.33. The set $T(\Gamma)$ should be thought of as the analogue of a droplet $T(\sigma_f)$ (this analogy will become transparent in Proposition 3.2). On the other hand, the notation $\Pi(\Gamma)$ is supposed to remind the readers that (the closure of) the unbounded component of $\mathcal{F}_{\Gamma}$ is a "polygon." <span id="page-11-0"></span>  Figure 3. Left: The circles Cⁱ generate a Kleinian reflection group Γ ∈ ∂β(Γ4). The map ρ^Γ is defined piece-wise on the union of the closed disks int Cⁱ . The fundamental domain F = F^Γ (for the action of Γ on Ω(Γ)) is the complement of these open disks with the singular boundary points removed. The connected components of F are marked. Right: The unbounded component of the domain of discontinuity Ω(Γ) is Ω∞(Γ). Every point in Ω(Γ) escapes to F under iterates of ρΓ. The point of tangential intersection of C² and C⁴ is the fixed point of an accidental parabolic of the index two Kleinian subgroup ^Γe. <span id="page-11-2"></span>Proposition 2.34. Let Γ ∈ β(Γd). Then: $$\Omega(\Gamma) = \bigcup_{n\geq 0} \rho_{\Gamma}^{-n}(\mathcal{F}_{\Gamma}), \text{ and } \Omega_{\infty}(\Gamma) = \bigcup_{n\geq 0} \rho_{\Gamma}^{-n}(\Pi^{o}(\Gamma)).$$ In particular, Λ(Γ) is completely invariant under ρΓ. Proof. This follows from Propositions [2.14,](#page-7-2) [2.28,](#page-9-1) [2.31.](#page-10-1) <span id="page-11-1"></span>Remark 2.35. Let Γ ∈ β(Γd). We now briefly describe the covering properties of ρ^Γ : Λ(Γ) → Λ(Γ). To this end, first note that $$\Lambda(\Gamma) = \bigcup_{i=1}^{d} \left( \overline{\operatorname{int} C_i} \cap \Lambda(\Gamma) \right)$$ (see Figure [3\)](#page-11-0). The interiors of these d "partition pieces" are disjoint, and ρ^Γ maps each of them injectively onto the union of the others. This produces a Markov partition for the degree d orientationreversing covering map ρ^Γ : Λ(Γ) → Λ(Γ). In the particular case of the base group Γd, the above discussion yields a Markov partition $$\mathbb{T} = \bigcup_{j=1}^{d} \left[ \exp\left(\frac{2\pi i (j-1)}{d}\right), \exp\left(\frac{2\pi i j}{d}\right) \right]$$ of the map $\rho_{\Gamma_d}: \mathbb{T} \to \mathbb{T}$ . Note that the expanding map $$\overline{z}^{d-1}: \mathbb{T} \to \mathbb{T}$$ . or equivalently, $$m_{-(d-1)}: \mathbb{R}/\mathbb{Z} \to \mathbb{R}/\mathbb{Z}, \ \theta \mapsto -(d-1)\theta$$ also admits the same Markov partition with the same transition matrix (identifying $\mathbb{T}$ with $\mathbb{R}/\mathbb{Z}$ ). Following [LLMM18a, §3.2], one can define a homeomorphism $$\mathcal{E}_{d-1}: \mathbb{T} \to \mathbb{T}$$ via the coding maps of $\rho_{\Gamma_d}|_{\mathbb{T}}$ and $\overline{z}^{d-1}|_{\mathbb{T}}$ such that $\mathcal{E}_{d-1}$ maps 1 to 1, and conjugates $\rho_{\Gamma_d}$ to $\overline{z}^{d-1}$ (or $m_{-(d-1)}$ ). Since both $\rho_{\Gamma_d}$ and $\overline{z}^{d-1}$ commute with the complex conjugation map and $\mathcal{E}_{d-1}$ fixes 1, one sees that $\mathcal{E}_{d-1}$ commutes with the complex conjugation map as well. The next result provides us with a model of the dynamics of $\rho_{\Gamma}$ on the limit set $\Lambda(\Gamma)$ as a quotient of the action of $\rho_{\Gamma_d}$ on the unit circle.

Proposition 2.36. Let $\Gamma \in \overline{\beta(\Gamma_d)}$ . There exists a conformal map $\phi_{\Gamma} : \mathbb{D}^* \to \Omega_{\infty}(\Gamma)$ such that <span id="page-12-1"></span>(3) $$\rho_{\Gamma_d}(z) = \phi_{\Gamma}^{-1} \circ \rho_{\Gamma} \circ \phi_{\Gamma}(z), \text{ for } z \in \mathbb{D}^* \setminus \operatorname{int} \Pi(\Gamma_d).$$ The map $\phi_{\Gamma}$ extends continuously to a semi-conjugacy $\phi_{\Gamma}: \mathbb{T} \to \Lambda(\Gamma)$ between $\rho_{\Gamma_d}|_{\mathbb{T}}$ and $\rho_{\Gamma}|_{\Lambda(\Gamma)}$ , and $\phi_{\Gamma}$ sends cusps of $\partial \Pi(\Gamma_d)$ to cusps of $\partial \Pi(\Gamma)$ with labels preserved. Proof. Recall that $\Gamma_d$ is generated by reflections $\rho_i$ in circles $(\mathbf{C}_i)_{i=1}^d$ . It follows from Propositions 2.14 and 2.28 that $\Pi^o(\mathbf{\Gamma}_d)$ and $\Pi^o(\Gamma)$ are fundamental domains for the actions of $\mathbf{\Gamma}_d$ and $\Gamma$ on $\mathbb{D}$ and $\Omega_\infty(\Gamma)$ , respectively. By the proofs of Lemma 2.22 and Proposition 2.23, there is a conformal mapping $\phi_\Gamma: \operatorname{int}\Pi(\mathbf{\Gamma}_d) \to \operatorname{int}\Pi(\Gamma)$ whose extension to $\partial\Pi(\mathbf{\Gamma}_d)$ is a label-preserving homeomorphism onto $\partial\Pi(\Gamma)$ . Thus, by the Schwarz reflection principle, we may extend $\phi_\Gamma$ to a conformal mapping $\phi_\Gamma: \mathbb{D}^ \to \Omega_\infty(\Gamma)$ which satisfies (3) by construction. Note that $\partial\Omega_{\infty}(\Gamma) = \Lambda(\Gamma)$ . By local connectedness of $\Lambda(\Gamma)$ (see Proposition 2.28), the map $\phi_{\Gamma}$ extends continuously to a semi-conjugacy $\mathbb{T} \to \Lambda(\Gamma)$ . We will now introduce the notion of a label-preserving homeomorphism, which will play an important role in the proof of Theorem A. <span id="page-12-2"></span>Remark 2.37. For $f_0(z):=z-1/(dz^d)$ , label the non-zero critical points of $f_0$ as $\xi_1^{f_0},\cdots,\xi_{d+1}^{f_0}$ in counter-clockwise order with $\xi_1^{f_0}=e^{\frac{i\pi}{d+1}}$ . Note that the critical points of f vary continuously depending on $f\in \Sigma_d$ , and $f\in \Sigma_d$ can not have a double critical point on $\mathbb{T}$ . Since $\Sigma_d$ is connected by Proposition 4.19, there is a unique labeling $\xi_1^f,\cdots,\xi_{d+1}^f$ of critical points of any $f\in \Sigma_d$ such that $f\mapsto \xi_i^f$ is continuous (for $i=1,\cdots,d+1$ ). This in turn determines a labeling of the cusps $\zeta_i^f:=f(\xi_i^f)$ of $f(\mathbb{T})$ such that $f\mapsto \zeta_i^f$ is continuous $(i=1,\cdots,d+1)$ . Similarly, label the cusps of $\partial T(\mathbf{\Gamma}_d)$ as $\eta_1,\cdots,\eta_d$ in counter-clockwise order with $\eta_1=1$ . This Similarly, label the cusps of $\partial T(\mathbf{\Gamma}_d)$ as $\eta_1, \dots, \eta_d$ in counter-clockwise order with $\eta_1 = 1$ . This determines a labeling of cusps of $\partial T(\Gamma)$ for any $\Gamma \in \overline{\beta(\mathbf{\Gamma}_d)}$ as the group $\Gamma$ is the image under a representation of $\mathbf{\Gamma}_d$ .

Theorem 2.44. [LV73, §I.4.9] If the sequence of quadrilaterals $Q_n$ converges to a quadrilateral Q, then $$\lim_{n\to\infty} M(Q_n) = M(Q).$$

Proposition 3.1. Let $\Gamma$ , $\Gamma'$ be necklace groups. Suppose there exist homeomorphisms $$h_1: T(\Gamma) \to T(\Gamma')$$ and $h_2: \Pi(\Gamma) \to \Pi(\Gamma')$ which agree on cusps of $\partial T(\Gamma)$ , and map cusps of $\partial T(\Gamma)$ to cusps of $\partial T(\Gamma')$ . Suppose furthermore that $h_1$ , $h_2$ are conformal on $T^o(\Gamma')$ , $F^o(\Gamma')$ , respectively. Then $h_1$ , $h_2$ are restrictions of a common $M \in \operatorname{Aut}(\mathbb{C})$ such that $$\Gamma' = M \circ \Gamma \circ M^{-1}.$$

Proposition 3.2. Let $f \in \Sigma_d^*$ . There exists a unique $\Gamma_f \in \overline{\beta(\Gamma_{d+1})}$ such that there is a label-preserving homeomorphism $$h: T(\Gamma_f) \to T(\sigma_f)$$ with h conformal on int $T(\Gamma_f)$ . Proof of Existence. We first assume that $f(\mathbb{T})$ has no double points. Let $$g: T(\mathbf{\Gamma}_{d+1}) \to T(\sigma_f)$$ be a label-preserving diffeomorphism such that $$||g_{\overline{z}}/g_z||_{L^{\infty}(T(\Gamma_{d+1}))} < 1.$$ Define a Beltrami coefficient $\mu_a$ by $$\mu_q(u) := g_{\overline{z}}(u)/g_z(u) \text{ for } u \in T(\Gamma_{d+1}),$$ and $$\mu_g(u) := \begin{cases} \mu_g(r_i^{\circ n}(u)) & \text{if } u \in r_i^{-n}(T(\mathbf{\Gamma}_{d+1})) \text{ for } 1 \leq i \leq d+1 \text{ and } n \geq 1, \\ 0 & \text{otherwise.} \end{cases}$$ Denote by $\tau_g:\widehat{\mathbb{C}}\to\widehat{\mathbb{C}}$ the integrating map of $\mu_g$ , normalized so that $$\tau_q(z) = z + O(1/|z|)$$ as $z \to \infty$ . We claim that $\Gamma_f := \tau_g \circ \mathbf{\Gamma}_{d+1} \circ \tau_g^{-1}$ satisfies the conclusions of Proposition 3.2. Indeed, $$\tau_g \circ \mathbf{\Gamma}_{d+1} \circ \tau_g^{-1} \in \beta(\mathbf{\Gamma}_{d+1})$$ since $\mu_q \equiv 0$ on $\mathbb{D}^*$ . The map $$h := g \circ \tau_g^{-1} : T(\Gamma_f) \to T(\sigma_f)$$ is conformal on int $T_{\Gamma_f}$ since $\tau_g$ is the integrating map for $g_{\overline{z}}/g_z$ . Lastly, we see that h is label-preserving since $\tau_g^{-1}$ and g are both label-preserving by definition. Next we consider the case that $f(\mathbb{T})$ has at least one double point. We claim the existence of $\Gamma \in \overline{\beta(\Gamma_{d+1})}$ such that there is a label-preserving diffeomorphism $g: T(\Gamma) \to T(\sigma_f)$ . Given the existence of such a $\Gamma$ , the same quasiconformal deformation argument as above produces the desired group $\Gamma_f$ and homeomorphism h. The existence of such a $\Gamma$ may be proven by pinching geodesics on the (d+1)-times punctured sphere $\mathbb{D}/\Gamma_d^+$ (where $\Gamma_d^+$ is the index 2 Kleinian subgroup of $\Gamma_{d+1}$ consisting of orientation-preserving automorphisms of $\mathbb{C}$ ), or adapting the techniques used in the proof of [LMM19, Theorem 4.11]. Alternatively, we may prove the existence of $\Gamma$ by associating a planar vertex $v_i$ , for $1 \leq i \leq d+1$ , to each analytic arc connecting two cusps of $f(\mathbb{T})$ , as in Figure 4. Connect two vertices $v_i$ , $v_j$ by an edge if and only if the corresponding analytic arcs have non-empty intersection. This defines a simplicial 2-complex K in the plane. K is a combinatorial closed disc, and hence [Ste05, Proposition 6.1] shows that there is a circle packing $(C'_i)_{i=1}^{d+1}$ of $\mathbb D$ for K, with each $C'_i$ tangent to $\partial \mathbb D$ . Quasiconformally deforming this circle packing group so that there is a label-preserving conformal map to $\Pi(\Gamma_{d+1})$ gives the desired $\Gamma \in \overline{\beta(\Gamma_{d+1})}$ (up to Möbius conjugacy). <span id="page-16-0"></span> FIGURE 4. Illustrated is the procedure of associating a circle packing to an element of $\Sigma_d^*$ . Proof of Uniqueness. If $\Gamma$ , $\Gamma'$ both satisfy the conclusions of the Proposition, we may take $h_1$ , $h_2$ as in Proposition 3.1, where $h_2(z) = z + O(1/z)$ as $z \to \infty$ since $\Gamma$ , $\Gamma' \in \overline{\beta(\Gamma_{d+1})}$ . Thus as $h_2$ extends to an automorphism of $\mathbb C$ by Proposition 3.1, it follows that $h_2 = \mathrm{id}$ , and hence $\Gamma = \Gamma'$ . Remark 3.3. The requirement that h be label-preserving is essential to the uniqueness statement in the conclusion of Proposition 3.2: see Figure 5.

Lemma 3.5. Let U, V be Jordan domains. Let $n \geq 4$ , and suppose $u_1, \dots, u_n \in \partial U$ and $v_1, \dots, v_n \in \partial V$ are oriented positively with respect to U, V (respectively). Suppose furthermore that the quadrilaterals $$U(u_i, u_{i+1}, u_k, u_{k+1}), V(v_i, v_{i+1}, v_k, v_{k+1})$$ have the same modulus for each j, k with $1 \le j < j+2 \le k \le n-1$ . Then there is a conformal map $$f: U \to V$$ such that $f(u_i) = v_i$ , $1 \le i \le n$ .

Lemma 4.1. Let $f \in \Sigma_d^*$ . Then <span id="page-20-2"></span><span id="page-20-1"></span> $$|\overline{\partial}\sigma_f(z)| > 1, \forall z \in f(\mathbb{D}^*).$$

Proposition 4.2. Let $f \in \Sigma_d^*$ . Then $\partial \mathcal{B}_{\infty}(\sigma_f)$ is locally connected. Remark 4.3. Our proof follows the strategy taken in [DH85, Chapter 10].

Proposition 4.4. Let $f \in \Sigma_d^*$ . Then $\partial \mathcal{B}_{\infty}(\sigma_f) = \partial \mathcal{T}_{\infty}(\sigma_f)$ . The proof of Proposition 4.4 will be carried out by way of several lemmas below. First we record the following definition:

Lemma 4.9. The landing points of the fixed external rays of $\sigma$ are singular points of $f(\mathbb{T})$ .

Lemma 4.10. int $\overline{\mathcal{T}_{\infty}(\sigma)} = \mathcal{T}_{\infty}(\sigma)$ . In particular, each component of $\mathcal{T}_{\infty}(\sigma)$ is a Jordan domain. Proof. Let $\mathcal{U}$ denote a component of $\mathcal{T}_{\infty}(\sigma)$ . We first show that $\partial \mathcal{U} \subset \partial \mathcal{B}_{\infty}(\sigma)$ . First assume $\mathcal{U}$ is the forward-invariant component of $\mathcal{T}_{\infty}(\sigma)$ which contains the landing point p of the 0-ray of $\sigma$ . As in the proof of Lemma 4.9, we note that $\sigma|_{\partial \mathcal{U}}$ is topologically semi-conjugate to $\rho_{\Gamma_{2+j_i}}|_{\mathbb{T}}$ . Thus the iterated pre-images of p under $\sigma$ are dense in $\partial \mathcal{U}$ . Since $p \in \partial \mathcal{B}_{\infty}(\sigma)$ by Lemma 4.9, and $\partial \mathcal{B}_{\infty}(\sigma)$ is completely invariant, it follows that $\partial \mathcal{U} \subset \partial \mathcal{B}_{\infty}(\sigma)$ . A similar argument applies to show that the boundary of any forward-invariant component of $\mathcal{T}_{\infty}(\sigma)$ is contained in $\partial \mathcal{B}_{\infty}(\sigma)$ . Lastly, any other component of $\mathcal{T}_{\infty}(\sigma)$ maps (under some iterate of $\sigma$ ) onto one of the invariant components of $\mathcal{T}_{\infty}(\sigma)$ , so that $\partial \mathcal{U} \subset \partial \mathcal{B}_{\infty}(\sigma)$ for any component $\mathcal{U}$ of $\mathcal{T}_{\infty}(\sigma)$ . Note that since $\mathcal{T}_{\infty}(\sigma)$ is open, we have $\mathcal{T}_{\infty}(\sigma) \subset \operatorname{int} \mathcal{T}_{\infty}(\sigma)$ . Let us now pick a component W of $\operatorname{int} \overline{\mathcal{T}_{\infty}(\sigma)}$ . Since $\partial \mathcal{T}_{\infty}(\sigma)$ is nowhere dense in $\mathbb{C}$ , it follows that W must intersect some component $\mathcal{U}$ of $\mathcal{T}_{\infty}(\sigma)$ . As W is a maximal open connected subset of $\overline{\mathcal{T}_{\infty}(\sigma)}$ , it follows $\mathcal{U} \subset W$ . However, if $\mathcal{U} \subsetneq W$ , then $\partial \mathcal{U}$ must contain some point not belonging to $\partial \overline{\mathcal{T}_{\infty}(\sigma)} = \partial \mathcal{B}_{\infty}(\sigma)$ , and this contradicts what was shown in the previous paragraph. Thus $\mathcal{U} = W$ , and so int $\overline{\mathcal{T}_{\infty}(\sigma)} \subset \mathcal{T}_{\infty}(\sigma)$ . The conclusion of the lemma follows. Proof of Proposition 4.4. Let $$x \in \partial \mathcal{T}_{\infty}(\sigma) \subset \overline{\mathcal{T}_{\infty}(\sigma)} = \left( \operatorname{int} \overline{\mathcal{T}_{\infty}(\sigma)} \right) \sqcup \partial \overline{\mathcal{T}_{\infty}(\sigma)}.$$ By Lemma 4.10 and the openness of $\mathcal{T}_{\infty}(\sigma)$ , it follows that $$x \in \partial \overline{\mathcal{T}_{\infty}(\sigma)} = \partial \mathcal{B}_{\infty}(\sigma).$$ Hence, $\partial \mathcal{T}_{\infty}(\sigma) \subset \partial \mathcal{B}_{\infty}(\sigma)$ , and together with Lemma 4.8, this proves Proposition 4.4. <span id="page-23-2"></span>Corollary 4.11. Let $f \in \Sigma_d^*$ . Then $$\widehat{\mathbb{C}} = \mathcal{B}_{\infty}(\sigma_f) \sqcup \Lambda(\sigma_f) \sqcup \mathcal{T}_{\infty}(\sigma_f),$$ where $\Lambda(\sigma_f) = \partial \mathcal{B}_{\infty}(\sigma_f) = \partial \mathcal{T}_{\infty}(\sigma_f)$ is the limit set of $\sigma_f$ .

Lemma 4.14. Let $f \in \Sigma_d^*$ . Then: - (1) Each cusp of $f(\mathbb{T})$ is the landing point of a unique external ray of $\mathcal{B}_{\infty}(\sigma_f)$ , and the angle of this ray is fixed under $m_{-d}$ . - (2) Each double point of $f(\mathbb{T})$ is the landing point of exactly two external rays of $\mathcal{B}_{\infty}(\sigma_f)$ , and the angles of the corresponding two rays form a 2-cycle under $m_{-d}$ .

Proposition 4.18. Let $f \in \Sigma_{d,k}^*$ , and $x \in \lambda(\sigma_f)$ be a non-trivial equivalence class. Then, the following hold true. - $\begin{array}{l} (1) \ |x|=2, \ and \ m_{-d}^{\circ n}(x)=\{\alpha_i,\alpha_i'\} \ for \ some \ n\geq 0 \ and \ 1\leq i\leq k. \\ (2) \ If \ n_0 \ is \ the \ smallest \ non-negative \ integer \ with \ m_{-d}^{\circ n_0}(x)=\{\alpha_i,\alpha_i'\}, \ then \ x \ is \ contained \ in \ a \\ connected \ component \ of \ \mathbb{T}\setminus m_{-d}^{-(n_0-1)}(\mathcal{A}^{double}\cup\mathcal{A}^{cusp}). \end{array}$ Proof of (1). We abbreviate $\sigma := \sigma_f$ . If $x \in \lambda(\sigma_f)$ is non-trivial, x is a collection of $\geq 2$ angles whose corresponding external rays for $\sigma$ land at a cut-point w of $\Lambda(\sigma)$ . Let $\theta$ , $\theta'$ be distinct angles in x. Suppose w is not a double point of $f(\mathbb{T})$ . As w is a cut-point of $\Lambda(\sigma)$ , it follows from Lemma 4.14 that w is not a cusp of $f(\mathbb{T})$ . Suppose by way of contradiction that no iterate of $\sigma$ maps w to a double point of $f(\mathbb{T})$ . Note that no iterate of $\sigma$ can map w to a cusp of $f(\mathbb{T})$ , as $\sigma$ is a local homeomorphism on $\Lambda(\sigma) \setminus f(\mathbb{T})$ and cusps of $f(\mathbb{T})$ are not cut-points of $\Lambda(\sigma)$ by Lemma 4.14. Thus, w has a well-defined itinerary (or symbol sequence) with respect to the Markov partition of $\Lambda(\sigma)$ in Remark 4.17. This implies that the angles $\theta$ and $\theta'$ have the same (well-defined) itinerary with respect to the corresponding Markov partition of $\mathbb{R}/\mathbb{Z}$ . However, this contradicts expansivity of the map $m_{-d}: \mathbb{R}/\mathbb{Z} \to \mathbb{R}/\mathbb{Z}$ , as the distance between $m_{-d}^{\circ j}(\theta)$ and $m_{-d}^{\circ j}(\theta')$ must exceed 1/(d+1) for some $j \in \mathbb{N}$ . This contradiction proves that $\sigma^{\circ n}(w)$ is a double point of $f(\mathbb{T})$ for some $n \geq 0$ . Let use choose the smallest n with this property, and call it $n_0$ . By Lemma 4.14, there are exactly two rays $\{\alpha_i, \alpha_i'\}$ landing at the double point $\sigma^{\circ n_0}(w)$ . As $\sigma$ is a local homeomorphism on $\Lambda(\sigma) \setminus f(\mathbb{T})$ , it follows that $\theta$ , $\theta'$ are the only two rays landing at w, and that $\sigma^{\circ n_0}$ maps the pair of rays at angles $\{\theta, \theta'\}$ to the pair of rays at angles $\{\alpha_i, \alpha_i'\}$ . In other words, $x = \{\theta, \theta'\}$ and $m_{-d}^{\circ n_0}(x) = \{\alpha_i, \alpha_i'\}$ . Proof of (2). This follows from the landing patterns of the rays corresponding to the angles in $\mathcal{A}^{\text{double}} \cup \mathcal{A}^{\text{cusp}}$ and injectivity of $\sigma$ on the interior of each piece of the Markov partition of $\Lambda(\sigma)$ defined in Remark 4.17. We conclude this subsection with a proof of connectedness of $\Sigma_d$ (which was used to define the labeling of the cusps on $f(\mathbb{T})$ in Remark 2.37), and a dynamical characterization of the cusp $\zeta_1^f$ as the landing point of the 0-ray of $\sigma_f$ , for $f \in \Sigma_d$ (which was used in the injectivity step of the proof of Proposition 3.4).

Proposition 4.19. $\Sigma_d^*$ is connected.

Proposition 4.20. Let $f_0(z) := z - 1/(dz^d)$ , $f \in \Sigma_d^*$ and $\omega_0 := e^{\frac{i\pi}{d+1}}$ . Then: - (1) The 0-ray of $\sigma_{f_0}$ lands at the cusp $\zeta_1^{f_0} = (1+1/d)\omega_0$ of $f_0(\mathbb{T})$ . - (2) The 0-ray of $\sigma_f$ lands at the cusp point $\zeta_1^f$ of $f(\mathbb{T})$ .

Proposition 4.24. Let $\Gamma \in \overline{\beta(\Gamma_d)}$ , and $x \in \lambda(\Gamma)$ be a non-trivial equivalence class. Then: - (1) |x|=2, and there is a double point $\eta$ of $\partial T(\Gamma)$ such that $\rho_{\Gamma_d}^{\circ n}(x)=\phi_{\Gamma}^{-1}(\eta)$ for some $n\geq 0$ . - (2) If $n_0$ is the smallest non-negative integer with the above property, then x is contained in a connected component of $\mathbb{T} \setminus \rho_{\Gamma_d}^{-(n_0-1)}(\Theta)$ . <span id="page-28-0"></span>Remark 4.25. For $f \in \Sigma_{d,k}^*$ with $k \geq 1$ , the index two Kleinian subgroup $\Gamma_f^+$ of $\Gamma_f$ has k accidental parabolics. These accidental parabolics correspond to a collection of k simple, closed, essential geodesics on $S^- = \mathbb{D}/\Gamma_d^+$ that can be pinched to obtain $\Gamma_f^+$ . These geodesics lift by $\Gamma_d$ to the universal cover $\mathbb{D}$ giving rise to a geodesic lamination of $\mathbb{D}$ [Mar07, §3.9]. By [MS13] (also compare [Mar07, p. 266]), the quotient of $\mathbb{T}$ by identifying the endpoints of the leaves of this lamination produces a topological model of the limit set $\Lambda(\Gamma_f^+) = \Lambda(\Gamma_f)$ . Therefore, up to rotation by a (d+1)-st root of unity, the set of equivalence classes of this geodesic lamination is equal to $\lambda(\Gamma_f)$ . Moreover, the continuous map $\phi_{\Gamma_f}: \mathbb{T} \to \Lambda(\Gamma_f)$ is a Cannon-Thurston map for $\Gamma_f$ (see [MS13, §2.2] for a discussion of Cannon-Thurston maps). <span id="page-28-2"></span>4.5. Relating The Laminations of Schwarz and Kleinian Limit Sets. Given $f \in \Sigma_d^*$ , we discussed the lamination of $\mathbb{T}$ induced by $\sigma_f$ in Subsection 4.3, and the lamination of $\mathbb{T}$ induced by $\Gamma_f$ in Subsection 4.4. The purpose of Subsection 4.5 is to relate these two laminations.

Proposition 4.26. Let $f \in \Sigma_d^*$ . Then the homeomorphism $\mathcal{E}_d : \mathbb{T} \to \mathbb{T}$ descends to a homeomorphism $$\mathcal{E}_d: \mathbb{T}/\lambda(\Gamma_f) \to \mathbb{T}/\lambda(\sigma_f).$$  FIGURE 7. A cartoon of the rays of period 1 and 2 landing on $\Lambda(\sigma)$ . Each cusp is the landing point of a unique fixed ray (in red), and each double point is the landing point of exactly two rays of period two (in green). The other rays of period two land at non-cut points of $\Lambda(\sigma)$ . These rays are colored such that the two rays of the same color form a 2-cycle. That the same pattern holds for the limit set $\Lambda(\Gamma)$ is the crux of the proof of Proposition 4.26.

Proposition 5.4. For each $f \in \Sigma_{d,k}^*$ , there exists an anti-polynomial $p_f$ of degree d such that: - (1) $p_f$ has a total of k+1 distinct critical points in $\mathbb{C}$ , - (2) Each critical point of $p_f$ is fixed by $p_f$ , and (3) The angled Hubbard tree of $p_f$ is isomorphic to $(\mathcal{T}(f), \deg, \angle)$ .

Proposition 5.5. Let $f \in \Sigma_{d,k}^*$ , and $p_f$ as in Proposition 5.4. Denote by $U_1, \dots, U_{k+1}$ the immediate attracting basins of the fixed critical points of $p_f$ . Then $$U := \bigcup_{i=1}^{k+1} \overline{U_i}$$ is connected. Moreover, $p_f$ has exactly 2k + d + 2 fixed points in $\mathbb{C}$ , of which: - (1) k+1 are critical points, - (2) k are cut-points of U and belong to $\mathcal{J}(p_f)$ , and - (3) d+1 are not cut-points of U and belong to $\mathcal{J}(p_f)$ .

Lemma 5.8. Let $f \in \Sigma_d^*$ , $p_f$ as in Proposition 5.4, and $\phi_p$ any Böttcher coordinate for $p_f$ . Then: - (1) Each $\beta \in \text{Rep}(p_f) \setminus \text{Cut}(p_f)$ is the landing point of a unique external ray. The angle of this external ray is fixed by $m_{-d}$ . - (2) Each $\beta \in \text{Rep}(p_f) \cap \text{Cut}(p_f)$ is the landing point of exactly two external rays. The angles of these two rays form a 2-cycle under $m_{-d}$ .

Proposition 5.9. Let $f \in \Sigma_d^*$ , and $p_f$ as in Proposition 5.4. There is a normalization of the Böttcher coordinate for $p_f$ such that $\lambda(p_f) = \lambda(\sigma_f)$ . Remark 5.10. The idea of the proof is similar to that of Proposition 4.26, for which we refer to Figure 8. Proof. Let k be such that $f \in \Sigma_{d,k}^*$ . We abbreviate $\sigma := \sigma_f$ , $p := p_f$ . Consider the isomorphism of the angled Hubbard tree of p with the abstract angled tree of f as defined in Example 5.3. Thus there is, first of all, a bijection between the attracting basins $U_1, \dots, U_{k+1}$ of p and the components $T_1, \dots, T_{k+1}$ of $T^o(\sigma)$ . We ensure the labeling is such that $U_i$ is mapped to $T_i$ . Since the deg function is preserved, the number of singular points on each $\partial T_i$ is equal to the number of fixed points of $p|_{\partial U_i}$ . Moreover, since the $\angle$ function is preserved, for each $1 \le i \le k+1$ there is a bijection $$\chi_i: \partial U_i \cap \operatorname{Rep}(p) \to \partial T_i \cap \operatorname{Sing}(f(\mathbb{T}))$$ satisfying: - (1) For $1 \leq j \leq k+1$ , one has $\chi_i(\beta) \in \partial T_i \cap \partial T_j$ if and only if $\beta \in \partial U_i \cap \partial U_j$ ; - (2) $(\beta_1, \beta_2, \beta_3)$ is oriented positively with respect to $U_i$ if and only if $(\chi_i(\beta_1), \chi_i(\beta_2), \chi_i(\beta_3))$ is oriented positively with respect to $T_i$ . By (1), the map $\chi : \text{Rep}(p) \to \text{Sing}(f(\mathbb{T}))$ defined piecewise as $\chi_i$ on each $\text{Rep}(p) \cap \partial U_i$ is well-defined, whence it follows that $\chi$ is a bijection. Denote by $\phi_p$ , $\phi_\sigma$ the Böttcher coordinates for p, $\sigma$ , respectively. We normalize $\phi_p$ so that $$\chi \circ \phi_n(1) = \phi_{\sigma}(1).$$ Recall that the cusps of $\partial T_i$ are the landing points of the fixed rays in $\partial \mathcal{B}_{\infty}(\sigma)$ by Lemma 4.14, and the points $\operatorname{Rep}(p) \setminus \operatorname{Cut}(p)$ are the landing points of the fixed rays in $\partial \mathcal{B}_{\infty}(p)$ by Lemma 5.8. The fixed rays of $\partial \mathcal{B}_{\infty}(\sigma)$ and $\partial \mathcal{B}_{\infty}(p)$ have the same angles, and we enumerate them $\theta_1, \dots, \theta_{d+1}$ where $\theta_1 := 0$ . There is a 2-cycle (under $m_{-d}$ ) on $\mathbb{T}$ in each pair of non-adjacent intervals $(\theta_i, \theta_{i+1})$ , $(\theta_j, \theta_{j+1})$ , and this constitutes all 2-cycles of $m_{-d}$ . By Lemma 5.8, for each $\beta \in \text{Rep}(p) \cap \text{Cut}(p)$ , the set $\phi_p^{-1}(\beta)$ is a 2-cycle on $\mathbb{T}$ . The 2-cycle $\phi_p^{-1}(\beta)$ is in the pair of intervals $(\theta_i, \theta_{i+1})$ , $(\theta_j, \theta_{j+1})$ if and only if $\beta$ lies on both $\phi_p((\theta_i, \theta_{i+1}))$ and $\phi_p((\theta_j, \theta_{j+1}))$ . Similarly, for each double point $\zeta$ of $f(\mathbb{T})$ , the set $\phi_\sigma^{-1}(\zeta)$ is a 2-cycle on $\mathbb{T}$ by Lemma 4.14. And moreover, the 2-cycle $\phi_\sigma^{-1}(\zeta)$ is in the pair of intervals $(\theta_i, \theta_{i+1})$ , $(\theta_j, \theta_{j+1})$ if and only if $\zeta$ lies on both $\phi_\sigma((\theta_i, \theta_{i+1}))$ and $\phi_\sigma((\theta_j, \theta_{j+1}))$ . Thus, by the definition of $\chi$ , for $\beta \in \text{Rep}(p) \cap \text{Cut}(p)$ , the 2-cycle $\phi_p^{-1}(\beta)$ is in the pair of intervals $(\theta_i, \theta_{i+1})$ , $(\theta_j, \theta_{j+1})$ if and only if $\phi_\sigma^{-1}(\chi(\beta))$ is in the same pair of intervals. As there is only one 2-cycle in any such pair, it follows that $\phi_p^{-1}(\beta) = \phi_\sigma^{-1}(\chi(\beta))$ . Recall that by Proposition 4.18, the pairs $\phi_{\sigma}^{-1}(\zeta)$ over all double points $\zeta$ of $f(\mathbb{T})$ generate $\lambda(\sigma)$ . A completely analogous proof to that of Proposition 4.18 shows $\lambda(p)$ is generated by pairs $\phi_p^{-1}(\beta)$ where $\beta$ ranges over $\operatorname{Rep}(p) \cap \operatorname{Cut}(p)$ . Thus since $\chi$ is a bijection and $\phi_p^{-1}(\beta) = \phi_{\sigma}^{-1}(\chi(\beta))$ for all $\beta \in \operatorname{Rep}(p) \cap \operatorname{Cut}(p)$ , it follows that $\lambda(\sigma) = \lambda(p)$ . <span id="page-36-0"></span>Remark 5.11. Let notation be as in Proposition 5.9, and denote by $\phi_{p_f}$ , $\phi_{\sigma_f}$ the Böttcher coordinates of $p_f$ , $\sigma_f$ (respectively) with $\phi_{p_f}$ normalized as in Proposition 5.9. It follows from Proposition 5.9 that $$\phi_{p_f} \circ \phi_{\sigma_f}^{-1} : \Lambda(\sigma_f) \to \mathcal{J}(p_f)$$ is well-defined, and indeed a topological conjugacy (see Figure 9). We note that Theorem B follows immediately from: <span id="page-37-3"></span> FIGURE 9. Various topological conjugacies. <span id="page-37-1"></span>Theorem C. Let $f \in \Sigma_d^*$ . Denote by $\sigma_f$ , $\Gamma_f$ , $p_f$ the Schwarz reflection map, Kleinian reflection group, and critically fixed anti-polynomial determined by Definition 2.3, Theorem A, and Proposition 5.4, respectively. Then the dynamical systems $$\sigma_f: \Lambda(\sigma_f) \to \Lambda(\sigma_f),$$ $$\rho_{\Gamma_f}: \Lambda(\Gamma_f) \to \Lambda(\Gamma_f),$$ $$p_f: \mathcal{J}(p_f) \to \mathcal{J}(p_f)$$ are pairwise topologically conjugate. Proof of Theorem C. That $\sigma_f|_{\Lambda(\sigma_f)}$ and $p_f|_{\mathcal{J}(p_f)}$ are topologically conjugate is a consequence of Proposition 5.9 as explained in Remark 5.11. That $\sigma_f|_{\Lambda(\sigma_f)}$ and $\rho_{\Gamma_f}|_{\Lambda(\Gamma_f)}$ are topologically conjugate follows from Proposition 4.26, as explained in Remark 4.27. Remark 5.12. In the spirit of [LLMM19, Theorem 7.2], it is natural to ask whether $\Lambda(\sigma_f)$ , $\Lambda(\Gamma_f)$ , $\mathcal{J}(p_f)$ can be distinguished by their quasisymmetry groups. Remark 5.13. In light of Proposition 5.9, we can conjugate $p_f$ by an affine map to assume that $p_f$ is monic, centered, and $\lambda(p_f) = \lambda(\sigma_f)$ , where $\lambda(p_f)$ is determined by the Böttcher coordinate of $p_f$ that is tangent to the identity at $\infty$ . In fact, $p_f$ becomes unique with such normalization. Moreover, it directly follows from the proof of Proposition 4.26 and Remark 4.25 that the circle homeomorphism $\mathcal{E}_d$ transports the geodesic lamination that produces a topological model for $\Lambda(\Gamma_f)$ to the lamination that produces a topological model for $\mathcal{J}(p_f)$ .

Definition 2.2. We will denote by $\Sigma_d^*$ the following class of rational maps: $$\Sigma_d^ := \left\{ f(z) = z + \frac{a_1}{z} + \dots + \frac{a_d}{z^d} : a_d = -\frac{1}{d} \text{ and } f|_{\mathbb{D}} \text{ is conformal.} \right\}.$$ Note that for each $d \geq 2$ , the space $\Sigma_d^*$ can be regarded as a slice of the space of schlicht functions: $$\Sigma := \left\{ f(z) = z + \frac{a_1}{z} + \dots + \frac{a_d}{z^d} + \dots : \ f|_{\mathbb{D}^*} \text{ is conformal} \right\}.$$ We endow $\Sigma_d$ with the topology of coefficient-wise convergence. Clearly, this topology is equivalent to that of uniform convergence on compact subsets of $\mathbb{D}$ .

Definition 2.3. Given $f \in \Sigma_d$ , we define the associated Schwarz reflection map $\sigma_f : f(\mathbb{D}^) \to \widehat{\mathbb{C}}$ by the following diagram: $$\mathbb{D}^* \xrightarrow{z \mapsto 1/\bar{z}} \mathbb{D}$$ $$f^{-1}$$ The map $\sigma_f : \sigma_f^{-1}(f(\mathbb{D}^)) \to f(\mathbb{D}^)$ is a proper branched covering map of degree d (branched only at $\infty$ ), and $\sigma_f : \sigma_f^{-1}(\operatorname{int} f(\mathbb{D}^)^c) \to \operatorname{int} f(\mathbb{D}^)^c$ is a degree (d+1) covering map. We also note that $\infty$ is a super-attracting fixed point of $\sigma_f$ ; more precisely, $\infty$ is a fixed critical point of $\sigma_f$ of multiplicity (d-1).

Definition 2.4. Let $f \in \Sigma_d^*$ . We define the basin of infinity for $\sigma_f$ as $$\mathcal{B}_{\infty}(\sigma_f) := \{ z \in \widehat{\mathbb{C}} : \sigma_f^{\circ n}(z) \xrightarrow{n \to \infty} \infty \}.$$ <span id="page-5-1"></span>Remark 2.5. Let $f \in \Sigma_d^*$ . Since $\sigma_f$ has no critical point other than $\infty$ in $\mathcal{B}_{\infty}(\sigma_f)$ , the proof of [Mil06, Theorem 9.3] may be adapted to show the existence of a Böttcher coordinate for $\sigma_f$ : a conformal map (1) $$\phi_{\sigma_f}: \mathbb{D}^ \to \mathcal{B}_{\infty}(\sigma_f) \text{ such that } \phi_{\sigma_f}^{-1} \circ \sigma_f \circ \phi_{\sigma_f}(u) = \overline{u}^d, \ \forall \ u \in \mathbb{D}^.$$ Since <span id="page-5-2"></span> $$\sigma_f(z) = -\frac{\overline{z}^d}{d} + O(\overline{z}^{d-1}) \text{ as } z \to \infty,$$ we may choose $\phi_{\sigma_f}$ such that <span id="page-5-0"></span>(2) $$\phi'_{\sigma_f}(\infty) = d^{\frac{1}{d-1}} e^{\frac{i\pi}{d+1}}.$$ As in [Mil06, Theorem 9.3], any Böttcher coordinate for $\sigma_f$ is unique up to multiplication by a $d+1^{\text{st}}$ root of unity. Thus, (2) determines a unique Böttcher coordinate $\phi_{\sigma_f}$ which we will henceforth refer to as the Böttcher coordinate for $\sigma_f$ . The set $\widehat{\mathbb{C}} \setminus f(\mathbb{D}^)$ is called the droplet, or fundamental tile, and is denoted by $T(\sigma_f)$ . By [LMM19, Proposition 2.8] and [LM14, Lemma 2.4], the curve $\partial T = f(\mathbb{T})$ has (d+1) distinct cusps and at most (d-2) double points. The desingularized droplet* $T^o(\sigma_f)$ is defined as $$T^o(\sigma_f) := T(\sigma_f) \setminus \{\zeta : \zeta \text{ is a cusp or double point of } f(\mathbb{T})\}.$$

Definition 2.6. The tiling set $\mathcal{T}_{\infty}(\sigma_f)$ is defined as: $$\mathcal{T}_{\infty}(\sigma_f) := T^o(\sigma_f) \cup \left\{ z \in \widehat{\mathbb{C}} : \sigma_f^{\circ n}(z) \in T^o(\sigma_f) \text{ for some } n \ge 1 \right\}.$$ Lastly, we define the limit set of $\sigma_f$ by $\Lambda(\sigma_f) := \partial \mathcal{T}_{\infty}(\sigma_f)$ . For more details on the space $\Sigma_d^*$ and the associated Schwarz reflection maps, we refer the readers to [LMM19].

Definition 2.8. A discrete subgroup $\Gamma$ of $\operatorname{Aut}^{\pm}(\widehat{\mathbb{C}})$ is called a Kleinian reflection group if $\Gamma$ is generated by reflections in finitely many Euclidean circles. <span id="page-6-2"></span>Remark 2.9. For a Euclidean circle C, consider the upper hemisphere $S \subset \mathbb{H}^3 := \{(x,y,t) \in \mathbb{R}^3 : t > 0\}$ such that $\partial S \cap \partial \mathbb{H}^3 = C$ . Reflection in the Euclidean circle C extends naturally to reflection in S, and defines an orientation-reversing isometry of $\mathbb{H}^3$ . Hence, a Kleinian reflection $\Gamma$ group can be thought of as a 3-dimensional hyperbolic reflection group. Since a Kleinian reflection group is discrete, by [VS93, Part II, Chapter 5, Proposition 1.4], we can choose its generators to be reflections in Euclidean circles $C_1, \dots, C_d$ such that: (\) For each i, the closure of the bounded component of $\widehat{\mathbb{C}} \setminus C_i$ does not contain any other $C_j$ . We will always assume that a chosen generating set for a Kleinian reflection group $\Gamma$ satisfies Conditions (\).

Definition 2.10. Let $\Gamma$ be a Kleinian reflection group. The domain of discontinuity of $\Gamma$ , denoted $\Omega(\Gamma)$ , is the maximal open subset of $\widehat{\mathbb{C}}$ on which the elements of $\Gamma$ form a normal family. The limit set of $\Gamma$ , denoted by $\Lambda(\Gamma)$ , is defined by $\Lambda(\Gamma) := \widehat{\mathbb{C}} \setminus \Omega(\Gamma)$ . <span id="page-6-1"></span> FIGURE 2. On the left is an interior necklace group, but the group generated by the circles pictured on the right violates condition (2) of Definition 2.11. For a Euclidean circle C, the bounded complementary component of C will be called the interior of C, and will be denoted by int C.

Definition 2.11. Let $\Gamma$ be a Kleinian reflection group. We say $\Gamma$ is a necklace group (see Figure 2) if it can be generated by reflections in Euclidean circles $C_1, \dots, C_d$ such that: - (1) each circle $C_i$ is tangent to $C_{i+1}$ (with i+1 taken mod d), - (2) the boundary of the unbounded component of $\widehat{\mathbb{C}} \setminus \bigcup_i C_i$ intersects each $C_i$ , and - (3) the circles $C_i$ have pairwise disjoint interiors. If, furthermore, $C_{i-1}$ and $C_{i+1}$ are the only circles to which any $C_i$ is tangent, then $\Gamma$ is an interior necklace group. Remark 2.12. In Definition 2.11, Condition (2) ensures that each circle $C_i$ is "seen" from $\infty$ - see Figure 2. When choosing a generating set for a necklace group, we always assume the generating set is chosen so as to satisfy Conditions (1)-(3), and the circles $C_1, \dots, C_d$ are labelled clockwise around $\infty$ . We note that a necklace group $\Gamma$ generated by reflections in d Euclidean circles is isomorphic to the free product of d copies of $\mathbb{Z}/2\mathbb{Z}$ . Notation 2.13. Given a necklace group $\Gamma$ with generating set given by reflections in circles $C_1, \dots, C_d$ , let $$\mathcal{F}_{\Gamma} := \widehat{\mathbb{C}} \setminus \left( \bigcup_{i=1}^{d} (\operatorname{int} C_i \cup \{C_j \cap C_i : j \neq i\}) \right).$$

Definition 2.15. Consider the Euclidean circles $\mathbf{C}_1, \dots, \mathbf{C}_d$ where $\mathbf{C}_j$ intersects |z| = 1 at right-angles at the roots of unity $\exp\left(\frac{2\pi i \cdot (j-1)}{d}\right)$ , $\exp\left(\frac{2\pi i \cdot j}{d}\right)$ . Let $\rho_j$ be the reflection map in the circle $\mathbf{C}_j$ . By [VS93, Part II, Chapter 5, Theorem 1.2], this defines a necklace group $$\Gamma_d := \langle \rho_1, \cdots, \rho_d : \rho_1^2 = \cdots = \rho_d^2 = 1 \rangle,$$ that acts on the Riemann sphere.

Definition 2.16. Let $\Gamma$ be a discrete subgroup of $\operatorname{Aut}^{\pm}(\widehat{\mathbb{C}})$ . An isomorphism $$\xi: \mathbf{\Gamma}_d \to \Gamma$$ is said to be weakly type-preserving, or w.t.p., if - (1) $\xi(g)$ is orientation-preserving if and only if g is orientation-preserving, and - (2) $\xi(g) \in \Gamma$ is a parabolic Möbius map for each parabolic Möbius map $g \in \Gamma_d$ . In order to construct the Bers slice of the group $\Gamma_d$ and describe its compactification, we need to define a representation space for $\Gamma_d$ . For necklace groups, the information encoded by a representation (defined below) is equivalent to the data given by a labeling of the underlying circle packing. We will see in Section 3 that working with the space of representations (as opposed to the space of necklace groups without a labeling of the underlying circle packings) is crucial for the homeomorphism statement of Theorem A (compare Figure 5).

Definition 2.17. We define $\mathcal{D}(\mathbf{\Gamma}_d) := \{ \xi : \mathbf{\Gamma}_d \to \Gamma | \Gamma \text{ is a discrete subgroup of } \mathrm{Aut}^{\pm}(\widehat{\mathbb{C}}), \text{ and } \xi \text{ is a w.t.p. isomorphism} \}.$ We endow $\mathcal{D}(\mathbf{\Gamma}_d)$ with the topology of algebraic convergence: we say that a sequence $(\xi_n)_{n=1}^{\infty} \subset \mathcal{D}(\mathbf{\Gamma}_d)$ converges to $\xi \in \mathcal{D}(\mathbf{\Gamma}_d)$ if $\xi_n(\rho_i) \to \xi(\rho_i)$ coefficient-wise (as $n \to \infty$ ) for $i = 1, \dots, d$ . Remark 2.18. Let $\xi \in \mathcal{D}(\Gamma_d)$ . Since for each $i \in \mathbb{Z}/d\mathbb{Z}$ , the Möbius map $\rho_i \circ \rho_{i+1}$ is parabolic (this follows from the fact that each $\mathbf{C}_i$ is tangent to $\mathbf{C}_{i+1}$ ), the w.t.p. condition implies that $\xi(\rho_i) \circ \xi(\rho_{i+1})$ is also parabolic. As each $\xi(\rho_i)$ is an anti-conformal involution, it follows that $\xi(\rho_i)$ is Möbius conjugate to the circular reflection $z \mapsto 1/\overline{z}$ or the antipodal map $z \mapsto -1/\overline{z}$ . A straightforward computation shows that the composition of $-1/\overline{z}$ with either the reflection or the antipodal map with respect to any circle has two distinct fixed points in $\widehat{\mathbb{C}}$ , and hence not parabolic. Therefore, it follows that no $\xi(\rho_i)$ is Möbius conjugate to the antipodal map $-1/\overline{z}$ . Hence, each $\xi(\rho_i)$ must be the reflection in some Euclidean circle $C_i$ . Thus, $\Gamma = \xi(\Gamma_d)$ is generated by reflections in the circles $C_1, \dots, C_d$ . The fact that $\xi(\rho_i) \circ \xi(\rho_{i+1})$ is parabolic now translates to the condition that each $C_i$ is tangent to $C_{i+1}$ (for $i \in \mathbb{Z}/d\mathbb{Z}$ ). However, new tangencies among the circles $C_i$ may arise. Moreover, that $\xi$ is an isomorphism rules out non-tangential intersection between circles $C_i$ , $C_j$ (indeed, a non-tangential intersection between $C_i$ and $C_j$ would introduce a new relation between $\xi(\rho_i)$ and $\xi(\rho_j)$ , compare [VS93, Part II, Chapter 5, §1.1]). Therefore, $\Gamma = \xi(\Gamma_d)$ is a Kleinian reflection group satisfying properties (1) and (3) of necklace groups.

Definition 2.19. Let $\tau$ be a conformal map defined in a neighborhood of $\infty$ with $\tau(\infty) = \infty$ . We will say $\tau$ is tangent to the identity at $\infty$ if $\tau'(\infty) = 1$ . We will say $\tau$ is hydrodynamically normalized if $$\tau(z) = z + O(1/z)$$ as $z \to \infty$ .

Definition 2.20. Let $\operatorname{Bel}_{\Gamma_d}$ denote those Beltrami coefficients $\mu$ invariant under $\Gamma_d$ , satisfying $\mu = 0$ a.e. on $\mathbb{D}$ . Let $\tau_{\mu} : \mathbb{C} \to \mathbb{C}$ denote the quasiconformal integrating map of $\mu$ , with the hydrodynamical normalization. The Bers slice* of $\Gamma$ is defined as $$\beta(\Gamma_d) := \{ \xi \in \mathcal{D}(\Gamma_d) \mid \xi(g) = \tau_\mu \circ g \circ \tau_\mu^{-1} \text{ for all } g \in \Gamma_d, \text{ where } \mu \in \mathrm{Bel}_{\Gamma_d} \}.$$ Remark 2.21. There is a natural free $\mathrm{PSL}_2(\mathbb{C})$ -action on $\mathcal{D}(\mathbf{\Gamma}_d)$ given by conjugation, and so it is natural to consider the space $\mathrm{AH}(\mathbf{\Gamma}_d) := \mathcal{D}(\mathbf{\Gamma}_d)/\mathrm{PSL}_2(\mathbb{C})$ . The following definition of the Bers slice, where no normalization for $\tau_{\mu}$ is specified, is more aligned with the classical Kleinian group literature: <span id="page-8-1"></span> $$\{\xi \in \mathrm{AH}(\Gamma_d) \mid \xi(g) = \tau_\mu \circ g \circ \tau_\mu^{-1} \text{ for all } g \in \Gamma_d, \text{ where } \mu \in \mathrm{Bel}_{\Gamma_d} \}.$$ Our Definition 2.20 of $\beta(\Gamma_d)$ is simply a canonical choice of representative from each equivalence class of $(\star)$ , and will be more appropriate for the present work.

Definition 2.24. We refer to $\overline{\beta(\Gamma_d)} \subset \mathcal{D}(\Gamma_d)$ as the Bers compactification of the Bers slice $\beta(\Gamma_d)$ . We refer to $\overline{\beta(\Gamma_d)} \setminus \beta(\Gamma_d)$ as the Bers boundary. Remark 2.25. We will often identify $\xi \in \overline{\beta(\Gamma_d)}$ with the group $\Gamma := \xi(\Gamma_d)$ , and simply write $\Gamma \in \overline{\beta(\Gamma_d)}$ , but always with the understanding of an associated representation $\xi : \Gamma_d \to \Gamma$ . Since $\xi$ is completely determined by its action on the generators $\rho_1, \dots, \rho_d$ of $\Gamma_d$ , this is equivalent to remembering the 'labeled' circle packing $C_1, \dots, C_d$ , where $\xi(\rho_i)$ is reflection in the circle $C_i$ , for $i = 1, \dots, d$ . Remark 2.26. The Apollonian gasket reflection group (see the right-hand side of Figure 2) is an example of a Kleinian reflection group in $\mathcal{D}(\mathbf{\Gamma}_d) \setminus \overline{\beta(\mathbf{\Gamma}_d)}$ . Notation 2.27. For $\Gamma \in \overline{\beta(\Gamma_d)}$ , we denote the component of $\Omega(\Gamma)$ containing $\infty$ by $\Omega_{\infty}(\Gamma)$ .

Definition 2.29. Let $\Gamma \in \overline{\beta(\Gamma_d)}$ , generated by reflections $(r_i)_{i=1}^d$ in circles $(C_i)_{i=1}^d$ . We define the associated reflection map $\rho_{\Gamma}$ by: $$\rho_{\Gamma}: \bigcup_{i=1}^{d} \overline{\operatorname{int}(C_{i})} \to \widehat{\mathbb{C}}$$ $$z \longmapsto r_{i}(z) \text{ if } z \in \overline{\operatorname{int}(C_{i})}.$$

Definition 2.30. Let $\Gamma$ be a Kleinian reflection group, and $f: D \to \widehat{\mathbb{C}}$ a mapping defined on a domain D. We say that $\Gamma$ and f are orbit-equivalent if for any two points $z, w \in \widehat{\mathbb{C}}$ , there exists $g \in \Gamma$ with g(z) = w if and only if there exist non-negative integers $n_1, n_2$ such that $f^{\circ n_1}(z) = f^{\circ n_2}(w)$ .

Definition 2.38. Let $f \in \Sigma_d^*$ and $\Gamma \in \overline{\beta(\Gamma_{d+1})}$ . We say that a homeomorphism $h: T(\Gamma) \to T(\sigma_f)$ is label-preserving if h maps cusps of $\partial T(\Gamma)$ to cusps of $\partial T(\sigma_f)$ , and h preserves the labeling of cusps of $\partial T(\Gamma)$ and $\partial T(\sigma_f)$ . Similarly, for $f, f' \in \Sigma_d$ (respectively, for $\Gamma, \Gamma' \in \overline{\beta(\Gamma_{d+1})}$ ), a homeomorphism $h: T(\sigma_f) \to T(\sigma_{f'})$ (respectively, $h: T(\Gamma) \to T(\Gamma')$ ) is called label-preserving* if h maps the boundary cusps to the boundary cusps preserving their labels. We conclude this subsection with a discussion of the connection between the Bers slice of the reflection group $\Gamma_d$ and a classical Teichmüller space. Let $\Gamma_d^+$ be the index two subgroup of $\Gamma_d$ consisting of all Möbius maps in $\Gamma_d$ . Then, $\Gamma_d^+$ is Fuchsian group (it preserves $\mathbb{D}$ and $\mathbb{D}$ ). Using Proposition 2.14, it is seen that the top and bottom surfaces $S^+ := \mathbb{D}^/\Gamma_d^+$ and $S^- := \mathbb{D}/\Gamma_d^+$ associated with the Fuchsian group $\Gamma_d^+$ are d times punctured spheres. Moreover, the anti-Möbius reflection $\rho_i$ in the circle $C_i$ descends to anti-conformal involutions on $S^\pm$ fixing all the punctures (the resulting involution is independent of $i \in \{1, \dots, d\}$ ). We will denote this involution on $S^-$ by $\iota$ . By definition, each $\xi \in \mathcal{D}(\Gamma_d)$ defines a discrete, faithful, w.t.p. representation of $\Gamma_d^+$ into $\mathrm{PSL}_2(\mathbb{C})$ . If $\xi \in \beta(\Gamma_d)$ , then $\xi$ is induced by a quasiconformal map that is conformal on $\mathbb{D}^*$ . Hence, such a representation of $\Gamma_d^+$ lies in the Bers slice of $\Gamma_d^+$ . Thus, $\beta(\Gamma_d)$ embeds into the Teichmüller space of a d times punctured sphere. On the other hand, each $\xi \in \beta(\Gamma_d) \setminus \beta(\Gamma_d)$ induces a representation of $\Gamma_d^+$ that lies on the boundary of the Bers slice of the Fuchsian group $\Gamma_d^+$ . The index two Kleinian group $\Gamma^+$ of $\Gamma := \xi(\Gamma_d)$ is geometrically finite (a fundamental polyhedron for the action of $\Gamma^+$ on $\mathbb{H}^3$ is obtained by "doubling" a fundamental polyhedron for $\Gamma$ , and hence it has finitely many sides). In fact, $\Gamma^+$ is a cusp group that is obtained by pinching a special collection of simple closed curves on $S^-$ . Indeed, since $S^-$ is equipped with a natural involution $\iota$ , any $\Gamma_d$ -invariant Beltrami coefficient on $\mathbb D$ induces an $\iota$ -invariant Beltrami coefficient on $S^-$ . Hence, the simple closed geodesics on $S^-$ that can be pinched via quasiconformal deformations with $\Gamma_d$ -invariant Beltrami coefficients are precisely the ones invariant under $\iota$ . Moreover, the $\iota$ -invariant simple closed geodesics on $S^-$ bijectively correspond to pairs of non-tangential circles $C_i$ and $C_j$ ; more precisely, they are the projections to $S^-$ of hyperbolic geodesics of $\mathbb D$ with end-points at the two fixed points of the loxodromic Möbius map $\rho_i \circ \rho_j$ . Hence, a group $\Gamma^+$ on the Bers boundary is obtained as a limit of a sequence of quasiFuchsian deformations of $\Gamma_d^+$ that pinch a disjoint union of $\iota$ -invariant simple, closed, essential geodesics on the bottom surface $S^-$ without changing the (marked) conformal equivalence class of the top surface $S^+$ . If $\xi(\rho_i)$ is reflection in the circle $C_i$ (for $i=1,\cdots,d$ ), then a point of intersection of some $C_i$ and $C_j$ with $j \neq i, i \pm 1 \pmod{d}$ corresponds to an accidental parabolic $\xi(\rho_i \circ \rho_j)$ for $\xi(\Gamma_d^+)$ . Furthermore, the quotient $$\mathcal{M}(\Gamma^+) := \left(\mathbb{H}^3 \cup \Omega(\Gamma^+)\right)/\Gamma^+$$ is an infinite volume 3-manifold whose conformal boundary $\partial \mathcal{M}(\Gamma^+) := \Omega(\Gamma^+)/\Gamma^+$ consists of finitely many punctured spheres. <span id="page-13-0"></span>2.3. Conformal Mating. In this Subsection we define the notion of conformal mating in Theorem A. Our definitions follow [PM12], to which we refer for a more extensive discussion of conformal mating. Notation 2.39. For $\Gamma \in \overline{\beta(\Gamma_d)}$ , recall $\Omega_{\infty}(\Gamma)$ denotes the unbounded component of $\Omega(\Gamma)$ . We let $\mathcal{K}(\Gamma) := \mathbb{C} \setminus \Omega_{\infty}(\Gamma)$ . <span id="page-14-1"></span>Remark 2.40. Let $w \mapsto p(w)$ be a monic, anti-holomorphic polynomial such that $\mathcal{J}(p)$ is connected and locally connected. Let $d := \deg(p)$ , and denote by $\phi_p : \mathbb{D}^* \to \mathcal{B}_{\infty}(p)$ the Böttcher coordinate for p such that $\phi'_p(\infty) = 1$ . We note that since $\partial \mathcal{K}(p) = \mathcal{J}(p)$ is locally connected by assumption, it follows that $\phi_p$ extends to a continuous semi-conjugacy between $z \mapsto \overline{z}^d|_{\mathbb{T}}$ and $p|_{\mathcal{J}(p)}$ . Now let $\Gamma \in \overline{\beta(\Gamma_{d+1})}$ . As was shown in Proposition 2.36, there is a natural continuous semi-conjugacy $\phi_{\Gamma} : \mathbb{T} \to \Lambda(\Gamma)$ between $\rho_{\Gamma_{d+1}}|_{\mathbb{T}}$ and $\rho_{\Gamma|\Lambda(\Gamma)}$ . Recall from Remark 2.35 that $\mathcal{E}_d : \mathbb{T} \to \mathbb{T}$ is a topological conjugacy between $\rho_{\Gamma_{d+1}}|_{\mathbb{T}}$ and $z \mapsto \overline{z}^d|_{\mathbb{T}}$ .

Definition 2.41. Let notation be as in Remark 2.40. We define an equivalence relation $\sim$ on $\mathcal{K}(\Gamma) \sqcup \mathcal{K}(p)$ by specifying $\sim$ is generated by $\phi_{\Gamma}(t) \sim \phi_{p}(\overline{\mathcal{E}_{d}(t)})$ for all $t \in \mathbb{T}$ .

Definition 2.42. Let $\Gamma \in \beta(\Gamma_{d+1})$ , p a monic, anti-holomorphic polynomial such that $\mathcal{J}(p)$ is connected and locally connected, and $f \in \Sigma_d$ . We say that $\sigma_f$ is a conformal mating* of $\Gamma$ with p if there exist continuous maps $$\psi_p: \mathcal{K}(p) \to \widehat{\mathbb{C}} \setminus \mathcal{T}_{\infty}(\sigma_f) \text{ and } \psi_{\Gamma}: \mathcal{K}(\Gamma) \to \overline{\mathcal{T}_{\infty}(\sigma_f)},$$ conformal on int $\mathcal{K}(p)$ , int $\mathcal{K}(\Gamma)$ , respectively, such that - (1) $\psi_p \circ p(w) = \sigma_f \circ \psi_p(w)$ for $w \in \mathcal{K}(p)$ , - (2) $\psi_{\Gamma}: T(\Gamma) \to T(\sigma_f)$ is label-preserving and $\psi_{\Gamma} \circ \rho_{\Gamma}(z) = \sigma_f \circ \psi_{\Gamma}(z)$ for $z \in \mathcal{K}(\Gamma) \setminus \operatorname{int} T^o(\Gamma)$ , - (3) $\psi_{\Gamma}(z) = \psi_{n}(w)$ if and only if $z \sim w$ where $\sim$ is as in Definition 2.41. - 2.4. Convergence of Quadrilaterals. We conclude Section 2 by recalling a notion of convergence for quadrilaterals (see [LV73, $\S$ I.4.9]) which will be useful to us in the proof of Theorem A. We will usually denote a topological quadrilateral by Q, and its modulus by M(Q).

Definition 2.43. The sequence of quadrilaterals $Q_n$ (with a-sides $a_i^n$ and b-sides $b_i^n$ , $i = 1, 2, n \in \mathbb{N}$ ) converges to the quadrilateral Q (with a-sides $a_i$ and b-sides $b_i$ , i = 1, 2) if to every $\varepsilon > 0$ there corresponds an $n_{\varepsilon}$ such that for $n \geq n_{\varepsilon}$ , every point of $a_i^n$ , $b_i^n$ , i = 1, 2, and every interior point of $Q_n$ has a spherical distance of at most $\varepsilon$ from $a_i$ , $b_i$ , and Q, respectively.

Definition 4.5. Let $f \in \Sigma_d^*$ . An external ray for $\sigma_f$ is a curve $$t \mapsto \phi_{\sigma_f}(te^{i\theta}), t \in (1, \infty)$$ for some $\theta \in [0, 2\pi)$ , where $\phi_{\sigma_f}$ is the Böttcher coordinate of Remark 2.5. For $\theta \in [0, 2\pi)$ , we refer to $$\{\phi_{\sigma_f}(te^{i\theta}): t \in (1,\infty)\}$$ as the $\theta$ -ray of $\sigma_f$ . Remark 4.6. By Proposition 4.2, each external ray of $\sigma_f$ lands, in other words $\lim_{t\to 1^+} \phi_{\sigma_f}(te^{i\theta})$ exists for each $\theta \in [0, 2\pi)$ . Notation 4.7. Let $\Sigma_{d,k}$ denote the collection of those $f \in \Sigma_d$ such that $f(\mathbb{T})$ has exactly k double points. For the remainder of this subsection we fix $f \in \Sigma_{d,k}^*$ and denote $\sigma := \sigma_f$ . Let us first record the straightforward inclusion: <span id="page-21-3"></span>Lemma 4.8. $\partial \mathcal{B}_{\infty}(\sigma) \subset \partial \mathcal{T}_{\infty}(\sigma)$ .

Definition 5.1. An abstract angled tree is a triple $(\mathcal{T}, \deg, \angle)$ , where: - (1) $\mathcal{T}$ is a tree, - (2) $\deg: V(\mathcal{T}) \to \mathbb{N}$ is a function with $\deg(v) \geq 2$ for each vertex v of $\mathcal{T}$ , - (3) valence(v) $\leq 1 + \deg(v)$ for each vertex v of $\mathcal{T}$ , and - (4) $\angle$ is a skew-symmetric, non-degenerate, additive function defined on pairs of edges incident at a common vertex, and takes values in $\frac{2\pi}{1+\deg(v)}\mathbb{Z}/2\pi\mathbb{Z}$ . Remark 5.2. If $(\mathcal{T}, \deg, \angle)$ is an abstract angled tree, the positive integer $$d := 1 + \sum_{v \in V(\mathcal{T})} (\deg(v) - 1)$$ is called the total degree of the angled tree. Two angled trees are said to be isomorphic if there is a tree isomorphism between them that preserves the functions deg and $\angle$ . <span id="page-32-3"></span>Example 5.3. To any $f \in \Sigma_{d,k}^*$ , we will associate an abstract angled tree $\mathcal{T}(f)$ with k+1 vertices as follows. Denote by $T_1, \dots, T_{k+1}$ the components of $T^o(\sigma_f)$ . Let $j_i \geq 0$ be such that the boundary of $T_i$ has $3 + j_i$ cusps. Assign a vertex $v_i$ to each component $T_i$ , and connect two vertices $v_i$ , $v_j$ by an edge if and only if $T_i$ , $T_i$ share a common boundary point. We define the deg function by: $$\deg: V(\mathcal{T}(f)) \to \mathbb{N}$$ $$v_i \mapsto 2 + j_i.$$ It remains to define the $\angle$ function for two edges e, e' meeting at a vertex $v_i$ . Suppose e, e' correspond to two cusps $\zeta$ , $\zeta' \in \partial T_i$ , and denote by $\gamma_i$ the component of $\partial T_i \setminus \{\zeta, \zeta'\}$ which, when traversed counter-clockwise, is oriented positively with respect to $T_i$ . Then $$\angle(e,e') := \frac{2\pi}{3+j_i} \cdot (1 + \#\{\text{cusps of } \partial T_i \text{ on the curve } \gamma_i\}).$$ We leave it to the reader to verify that $(\mathcal{T}(f), \deg, \angle)$ satisfies Definition 5.1 of an abstract angled tree. Note that if $f \in \Sigma_{d,d-2}^*$ , then the tree $(\mathcal{T}(f), \deg, \angle)$ is simply a bi-angled tree in the language of [LMM19, §2.5].