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Transcendental maps and Fuchsian groups unified in infinite-degree correspondences

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2026-07-08 16:46 UTC pith:IGWOW7YG

load-bearing objection First general construction of transcendental correspondences mating entire maps with Fuchsian groups; David integrability in the attracting case is the one soft spot to check. the 2 major comments →

arxiv 2607.06090 v1 pith:IGWOW7YG submitted 2026-07-07 math.DS math.DG

Transcendental correspondences: when Fuchsian groups take over basins of entire maps

Kostiantyn Drach , Sabyasachi Mukherjee , Leticia Pardo-Sim\'on This is my paper
classification math.DS math.DG
keywords holomorphic correspondencestranscendental dynamicsFuchsian groupsquasiconformal surgeryDavid surgerySullivan dictionaryfactor Bowen-Series mapsconformal welding
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs holomorphic correspondences of bi-degree (∞:∞) that simultaneously carry the dynamics of a transcendental entire function outside one of its Fatou basins and the action of a Fuchsian group inside that basin. This is the transcendental analogue of the algebraic correspondence theory that mates rational maps with Kleinian groups. The central mechanism is a two-step construction: first, a combination theorem (quasiconformal surgery in the parabolic case, David surgery in the attracting case) replaces the dynamics of the entire function f on a marked attracting or parabolic basin U with the boundary dynamics of a factor Bowen–Series map associated to a compatible Fuchsian group G, producing a partially defined holomorphic map g. Second, a conformal welding along the boundary where g acts as an orientation-reversing involution produces a meromorphic function h with a unique simple pole at the origin. The correspondence C is then defined by the equation [h(w) − h(η(z))]/[w − η(z)] = 0, where η(z) = 1/z is the Möbius involution. The result is that one forward branch of C reproduces f on its Julia set and outside the grand orbit of U, while the branches preserving a distinguished component of the expulsion set generate a group conjugate to G acting properly discontinuously. When f has finitely many singular values (Speiser class), the line complex of h can be described explicitly from the line complex of f and the combinatorial data of G.

Core claim

The paper proves that for every degree d ≥ 2, every entire map f in a suitable class (hyperbolic with quasidisk Fatou components, or strongly geometrically finite with parabolic basins) with a marked basin U of degree d, and every compatible Fuchsian group G of the same boundary degree d, there exists a meromorphic function h: ℂ → ℂ̂ with a unique simple pole at 0 such that the deleted covering correspondence of h composed with η(z) = 1/z yields a bi-degree (∞:∞) holomorphic correspondence that simultaneously realizes f outside the grand orbit of U and G inside a component of the expulsion set. The construction works in both attracting (David surgery) and parabolic (quasiconformal surgery)刚性

What carries the argument

The load-bearing objects are: (1) factor Bowen–Series maps, which encode the boundary action of a Fuchsian group as a continuous expansive circle covering map of degree d, providing the group-side input; (2) a combination theorem using quasiconformal surgery (parabolic case) or David surgery (attracting case) that replaces the dynamics of f on a Fatou basin with the factor Bowen–Series map, producing a partially defined holomorphic map g; (3) a conformal welding construction along the boundary where g is an orientation-reversing involution, which produces the meromorphic function h with a single simple pole; (4) the correspondence formula (z,w) ∈ C ⟺ [h(w) − h(η(z))]/[w − η(z)] = 0, which is

Load-bearing premise

In the attracting case, the David surgery requires that a certain Beltrami coefficient satisfies an exponential decay condition for large dilatation across all iterated preimage components of the basin. The proof verifies this using uniform quasidisk geometry and uniform degree bounds on preimage components; if these uniform geometric controls fail for some map in the class, the David integrability could break and the correspondence would not be well-defined.

What would settle it

Find a hyperbolic entire map f satisfying the stated class conditions (bounded quasidisk Fatou components, finitely many critical points per component) for which the pulled-back Beltrami coefficient in the David surgery fails the exponential decay condition (4.6) — i.e., the set of large dilatation does not shrink fast enough under iteration. This would invalidate the attracting-case combination theorem and hence the correspondence construction for that map.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The construction extends the Sullivan dictionary between rational dynamics and Kleinian groups to the transcendental setting, providing the first systematic family of transcendental correspondences beyond the single prior example of Bullett–Freiberger.
  • When f has finitely many singular values, the line complex of h is explicitly computable from the line complex of f and the orbifold data of G, giving a combinatorial handle on the resulting correspondence.
  • The construction admits extension to replacing dynamics on multiple Fatou components simultaneously, which would embed products of Teichmüller spaces of genus-zero orbifolds into spaces of transcendental correspondences.
  • The paper raises the question of pre-compactness and boundary behavior of families of such correspondences as the group varies in its Teichmüller space, paralleling degeneration results for algebraic correspondences.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the David integrability condition fails for some map f in the attracting class — for instance, if the uniform quasidisk geometry or uniform degree bounds on preimage components break down — the entire correspondence construction in the attracting case would collapse, leaving only the parabolic case as robust.
  • The correspondence's bi-degree (∞:∞) is intrinsic to the transcendental setting: the essential singularity of f at infinity forces h to be transcendental meromorphic, so no finite-degree algebraic correspondence can capture this mating. This suggests a structural barrier to any finite-degree approximation of these matings.
  • The explicit line complex construction for Speiser-class maps suggests that the combinatorial type of h is determined by a gluing of the Speiser graph of f with the orbifold signature of G, which could in principle be used to classify all such correspondences for a given singular-value count.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 7 minor

Summary. This paper initiates a systematic study of holomorphic correspondences of bi-degree (∞:∞) that arise as conformal matings of transcendental entire maps with Fuchsian groups. Given an entire map f in a class K_⋄(d) (attracting or parabolic) with a marked invariant basin U, and a compatible Fuchsian group G in G_⋄(d), the authors construct a holomorphic correspondence C ⊂ ℂ* × ℂ* that simultaneously (i) has a forward branch conformally conjugate to f outside the grand orbit of U, (ii) has branches generating a group conjugate to G acting on a component of the expulsion set, and (iii) is expressible as a deleted covering correspondence of a meromorphic function h with a unique simple pole at 0, composed with the Möbius involution η(z) = 1/z. The construction proceeds via quasiconformal surgery (parabolic case) or David surgery (attracting case) to build a conformal combination map g, followed by a welding construction to produce h, and finally a verification that the resulting correspondence C has the required dynamical properties. The paper also describes the line complex of h when f is in the Speiser class.

Significance. The paper extends the theory of algebraic correspondences (matings of rational maps with Fuchsian groups) to the transcendental setting, a direction previously explored only in the special case of [BF08]. The construction is parameter-free: given f and G, the correspondence C is determined by the surgery and welding, with no fitted constants. The meromorphic function h is constructed from the geometry of g via welding, not postulated. The parabolic case (§4.1) uses standard quasiconformal surgery cleanly. The attracting case (§4.2) involves a non-trivial David integrability estimate (Lemma 4.2) that carefully controls pullback geometry using uniform quasidisk properties (Proposition 2.8) and uniform degree bounds (Lemma 2.7). The line complex description in §5.1 for Speiser-class maps provides a concrete combinatorial handle on the output. The main theorem is a genuine extension of the Sullivan dictionary to the transcendental entire setting.

major comments (2)
  1. [§4.2, Lemma 4.2, proof (p. 24)] The key estimate in Lemma 4.2 for s=1 invokes [Zha16, Lemma 6.4] to obtain σ_Euc(B_i) ≤ C_1 · diam²_Euc(U_i) · e^{−α_1/ε} (equation (4.9)). The paper [Zha16] concerns PZ-type Siegel disks in the sine family, and the manuscript does not state the hypotheses of [Zha16, Lemma 6.4] or explain why it applies to general f ∈ K_attr(d). The estimate itself — exponential decay of the David bad set under pullback by a proper holomorphic map of bounded degree with controlled geometry — is plausible and should follow from the area formula combined with Koebe distortion and the η-bounded geometry of U_i (established via [CDKS22, Lemma A.7]). However, if the specific formulation in [Zha16] uses properties particular to sine maps (e.g., specific critical point structure or covering properties), the citation may not directly transfer. This is the single load-bearing point for the David integrability in
  2. [§4.2, Lemma 4.2, second claim (p. 25)] In the proof of the second claim of Lemma 4.2, the Koebe distortion argument yields the estimate σ((f^k|U'_i)^{-1}(E)) ≤ L · σ(U'_i) · σ(E)/σ(U_i) for Borel sets E ⊂ U_i. The application with E = B_i requires that B_i ⊂ U_i, which holds by construction. However, the uniformity of the constant L over all i ∈ I_N and all k ≥ 1 depends on the Koebe space mod(V'_i ∖ U'_i) being uniformly bounded below. The argument shows mod(V'_i ∖ U'_i) = mod(V_i ∖ U_i) ≥ (1/D) mod(V ∖ U), where D is the uniform degree bound for landing maps f^N: U_i → U. This step is correct, but the role of D should be made explicit: the uniform lower bound on the Koebe space is inversely proportional to D, and D is bounded by Lemma 2.7. A brief sentence clarifying that the Koebe constant L depends on D (and hence is uniform) would strengthen the argument.
minor comments (7)
  1. [§1.2, Main Theorem] The sets R₀, E, R are introduced in the Main Theorem statement before their definitions are given (the footnote helps, but the notation R vs R₀ could be clarified earlier). Consider adding a forward reference to §6.1 where these sets are formally defined.
  2. [§3.3] The notation H_d for the Hecke group with q = d+1 is introduced but the relationship between the parameter q and the degree d could be stated more prominently, as it is easy to miss that d = q−1.
  3. [§4.2, equation (4.5)] The Beltrami coefficient μ is defined using the notation (ψ∘φ∘f^k)^*(μ₀) for k ∈ ℕ_{≥1}. It would help to explicitly note that f^k here refers to the k-th iterate of f restricted to the appropriate preimage component, to avoid any ambiguity with the global iterate.
  4. [§5, Theorem 5.1] In the proof, the welding construction produces a Riemann surface Σ that is argued to be a punctured sphere. The argument that boundary components corresponding to parabolic fixed points reduce to points (via infinite modulus of the annulus Ã) is clean, but a citation for the standard result used (e.g., [Oik63, Theorem 1] is cited, but the criterion that infinite modulus implies removable boundary could be stated more explicitly) would aid the reader.
  5. [§6.3, Proposition 6.8] The proof of Proposition 6.8 is described as 'analogous to that of [MM25, Proposition 5.7]'. While the authors do work out details, the verification that the side-pairing transformations of T̃₀ correspond to the generators of G under Φ₀^{-1} is somewhat compressed. A brief expansion of this step would improve readability.
  6. [Figures 1–2] The figures are helpful. In Figure 2, the caption mentions G_attr(2) consists of two groups, but the text in §3.3 defines G_attr more generally. Ensure consistency between the figure caption and the general definition.
  7. [References] The reference [EFGP26] has a 2026 date; verify this is the correct publication year. Similarly for [BLLM26], [LM26], [LoM26], [LMM26].

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for a careful reading and for identifying two specific points in the proof of Lemma 4.2 that require clarification. Both points are well-taken and will be addressed in the revision. The first concerns the applicability of a citation to a different setting; we will replace the citation with a self-contained argument. The second concerns making explicit the dependence of a Koebe constant on the uniform degree bound. Neither point requires changes to the statements of any results, only to the exposition of the proof.

read point-by-point responses

  1. Referee: The key estimate in Lemma 4.2 for s=1 invokes [Zha16, Lemma 6.4] to obtain the exponential decay estimate (equation (4.9)). The referee notes that [Zha16] concerns PZ-type Siegel disks in the sine family, and questions whether the hypotheses of [Zha16, Lemma 6.4] transfer to general f in K_attr(d). The referee observes that the estimate should follow from the area formula, Koebe distortion, and eta-bounded geometry, but that the specific formulation in [Zha16] may use properties particular to sine maps.

    Authors: The referee is correct that the citation to [Zha16, Lemma 6.4] is not directly transferable as stated, since that result is formulated in the specific context of PZ-type Siegel disks for sine maps and may use properties particular to that family. We will revise the manuscript to replace the citation with a self-contained argument. Specifically, the estimate (4.9) — exponential decay of the David bad set under pullback by a proper holomorphic map of bounded degree with controlled geometry — follows from the area formula combined with Koebe distortion and the eta-bounded geometry of the preimage components U_i (established via [CDKS22, Lemma A.7]). The key ingredients are: (1) the eta-bounded geometry of U_i, which gives the diameter-area comparison (4.8); (2) the uniform degree bound from Lemma 2.7, which controls the branching of f: U_i -> U; and (3) the Koebe distortion theorem applied to the proper map f: V_i -> V, where V_i is the preimage component containing U_i. These ingredients are all established in the manuscript and are valid for general f in K_attr(d), not just sine maps. We will write out this argument explicitly in the revised proof, removing the reference to [Zha16]. revision: yes

  2. Referee: In the proof of the second claim of Lemma 4.2, the uniformity of the Koebe constant L over all i in I_N and all k >= 1 depends on the Koebe space mod(V'_i minus U'_i) being uniformly bounded below. The argument shows mod(V'_i minus U'_i) = mod(V_i minus U_i) >= (1/D) mod(V minus U), where D is the uniform degree bound for landing maps. The referee asks that the role of D be made explicit: the uniform lower bound on the Koebe space is inversely proportional to D, and D is bounded by Lemma 2.7.

    Authors: We agree that the role of D should be made explicit. The uniform lower bound on the Koebe space mod(V'_i minus U'_i) is indeed inversely proportional to D, and D is bounded by Lemma 2.7. We will add a clarifying sentence in the revised proof stating that the Koebe constant L depends on D (via the modulus inequality mod(V'_i minus U'_i) >= (1/D) mod(V minus U)), and since D is uniformly bounded by Lemma 2.7, the constant L is uniform over all i in I_N and all k >= 1. revision: yes

Circularity Check

0 steps flagged

No significant circularity; construction is parameter-free with independently verifiable external citations

full rationale

The paper constructs a holomorphic correspondence C from two inputs (an entire map f and a Fuchsian group G) via quasiconformal/David surgery and welding. The derivation chain is: (1) define a Beltrami coefficient μ from the geometry of f and the boundary conjugacy ψ (Eq. 4.5), (2) prove μ satisfies the David integrability condition (Claim I, Lemma 4.2), (3) integrate μ to obtain a David homeomorphism Ψ and define g = Ψ ∘ f̂ ∘ Ψ⁻¹ (Claim II), (4) construct a meromorphic h via welding on the boundary involution of g (Theorem 5.1), (5) define C from h and η (Eq. 6.2), and (6) verify the dynamical properties of C (Theorem 6.2). No step reduces to its inputs by definition. The correspondence C is not postulated and then shown to equal a fit; it is constructed from the surgery output g, and its properties are verified against the original f and G. Self-citations ([MM25, LLM24, BLLM26, LMMN25]) provide the factor Bowen–Series machinery and David extension results; these are independently established results with their own proofs, not ansätze smuggled in. The citation [Zha16, Lemma 6.4] used in Lemma 4.2 is an external result by a different author; whether it applies to general K_attr maps is a correctness concern, not a circularity issue. The line complex construction in §5.1 is illustrative, not load-bearing for the main theorem. The construction is parameter-free: given f and G, C is determined by the surgery and welding with no fitted constants. The only minor concern is that several load-bearing citations ([MM25], [LMMN25]) share co-authors with the present paper, but these results have independent proofs and are not merely restatements of the present paper's claims, so they do not create circularity.

Axiom & Free-Parameter Ledger

2 free parameters · 6 axioms · 3 invented entities

The paper introduces no ad-hoc free parameters or postulated entities. The classes K_⋄ and G_⋄ are defined by standard dynamical/geometric conditions with concrete examples. The meromorphic function h is constructed, not assumed. The axioms are all standard results (David integrability, quasiconformal surgery) or domain assumptions established in cited prior work (factor Bowen–Series properties, compatibility theorems, Fatou component geometry).

free parameters (2)
  • d (degree) = integer ≥ 2
    Input parameter specifying the degree of the Blaschke product on the map side and the boundary degree of the factor Bowen–Series map on the group side.
  • ⋄ (type) = attr or par
    Input parameter selecting attracting or parabolic regime, determining which surgery technique (David or quasiconformal) is used.
axioms (6)
  • standard math David Integrability Theorem: if μ satisfies the exponential decay condition (4.6), then the Beltrami equation has a David homeomorphism solution.
    Invoked in §4.2 (Claim I/Claim II) to integrate the Beltrami coefficient μ and obtain the straightening map Ψ. This is a known result from [AIM09].
  • standard math Quasiconformal surgery / Measurable Riemann Mapping Theorem
    Used in §4.1 (parabolic case) to integrate the Beltrami coefficient μ with bounded dilatation and obtain the straightening map Φ.
  • domain assumption Factor Bowen–Series maps are virtually mateable and orbit equivalent to the associated Fuchsian group
    Stated in Theorem 3.5, cited from [MM25, MM23]. The entire group-side construction depends on these properties.
  • domain assumption Compatibility theorems: boundary conjugacy between Blaschke products and factor Bowen–Series maps extend to David (attracting) or quasiconformal (parabolic) homeomorphisms of the disk
    Theorems 3.10 and 3.11, cited from [LMMN25, BLLM26]. These provide the boundary matching needed for the surgery in §4.
  • domain assumption Strongly geometrically finite entire maps have bounded Jordan domain Fatou components (Theorem 2.2) and hyperbolic entire maps in the specified class have uniform quasidisk Fatou components (Theorem 2.5, Proposition 2.8)
    These geometric control results are essential for the surgery and are cited from [ARS22, BFR15].
  • domain assumption Welding via orientation-reversing involutions produces a genus-zero Riemann surface (punctured sphere) when the local dynamics near singular boundary points yields infinite-modulus annuli
    Used in Theorem 5.1 to conclude that the welded surface Σ is a punctured sphere, based on [Oik63].
invented entities (3)
  • Classes K_⋄(d) of mateable entire maps independent evidence
    purpose: Define the map-side input: transcendental entire functions with controlled Fatou component geometry and marked attracting/parabolic basins of specified degree.
    These classes are defined by standard dynamical conditions (geometrically finite, hyperbolic, bounded Fatou components) that are well-studied in the literature. The paper provides concrete examples (sin z, (π/2)sin z).
  • Classes G_⋄(d) of compatible Fuchsian groups independent evidence
    purpose: Define the group-side input: Fuchsian groups associated to genus-zero orbifolds with specific combinatorial data, equipped with factor Bowen–Series maps.
    These are standard Fuchsian groups (Hecke groups, modular group, punctured sphere groups) with well-understood geometry. The factor Bowen–Series machinery is established in cited prior work.
  • The meromorphic function h with unique simple pole independent evidence
    purpose: Provides the analytic realization of the correspondence C via the deleted covering formula (1.1).
    h is constructed via welding from the conformal combination g, not postulated. Its existence follows from the surgery and uniformization. Its line complex can be explicitly described when f has finitely many singular values (§5.1).

pith-pipeline@v1.1.0-glm · 40278 in / 3327 out tokens · 304563 ms · 2026-07-08T16:46:55.050091+00:00 · methodology

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read the original abstract

In this paper, we initiate a systematic study of $(\infty : \infty)$ holomorphic correspondences that naturally arise as conformal combinations (matings) of transcendental entire maps with Fuchsian groups. This construction parallels the recent theory of finite-degree algebraic correspondences associated with rational maps. Our correspondence combines the dynamics of a transcendental entire function outside a distinguished attracting/parabolic basin with the action of a compatible Fuchsian group within it. We show that the resulting correspondence is the composition of a M\"obius involution and the deleted covering correspondence of a meromorphic function having exactly one simple pole. When the transcendental entire function has finitely many singular values, so does this meromorphic function, and its line complex can be described explicitly.

Figures

Figures reproduced from arXiv: 2607.06090 by Kostiantyn Drach, Leticia Pardo-Sim\'on, Sabyasachi Mukherjee.

Figure 1
Figure 1. Figure 1: The degree-two examples on the entire-map side. Top: fpzq “ sin z P Kparp2q. The unique fixed point 0 is parabolic and has two immediate parabolic basins, shown in black and gray. Bottom: fpzq “ π 2 sin z P Kattrp2q; the unique fixed points ˘π{2 are superattracting, and their immediate basins are shown in black and gray. In both examples, the Fatou set consists precisely of the grand orbits of the two indi… view at source ↗
Figure 2
Figure 2. Figure 2: Depicted are tessellations of the unit disk under the actions of the modular group (left) and the Fuchsian group uniformizing a sphere with two punctures and one orbifold point of order 2 (right). These are precisely the two groups in Gattrp2q. On the other hand, Gparp2q consists only of the modular group. immediate parabolic basin) with the action of the group (see [LLM24, BLLM26]). The situation for tran… view at source ↗
Figure 3
Figure 3. Figure 3: Construction of a factor Bowen–Series map in the fully ramified Hecke example with n “ 4 and p “ 1. Left: the preferred fundamental domain Π for the covering orbifold Σ and the associated Bowen–Series map r ABS Γ . The four sides of Π are paired with themselves by the transformations h1, . . . , h4, obtained from one another by rotation through powers of Mipzq “ iz. The blue arrows indicate the correspondi… view at source ↗
Figure 4
Figure 4. Figure 4: The local dynamics of a factor Bowen–Series map near a parabolic fixed point is depicted. The parabolic dynamics of F at x implies that there are inverse orbits ¨ ¨ ¨ FÝÑ b´n FÝÑ b´pn´1q FÝÑ ¨ ¨ ¨ FÝÑ b´1 FÝÑ b0 [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The local dynamics of a factor Bowen–Series map near a parabolic 2´cycle is depicted. arcs > b 1 0 , a´1 and c >1 0 , d´1) are contained in the connected component U of Dzγ having x on its boundary (respectively, the connected component U 1 of Dzγ 1 having Fpxq on its boundary). We also denote the F´preimages of b 1 0 , c1 0 on the arcs > b0, a1 ´1 and c > 0, d1 ´1 by b 1 ´1 , c1 ´1 . Similarly, we denote … view at source ↗
Figure 6
Figure 6. Figure 6: Illustrated is the welding construction of Theorem 5.1. m Define a meromorphic map h : Σ Ñ Cp as h ” φ´ on Σ´ and h ” g ˝ φ` on Σ ` z tφ ´1 ` p8qu. The fact that the piecewise definitions agree along BΣ ` “ BΣ ´ fol￾lows from the fact that the welding was done via the map g; i.e., from the relation φ` “ g ˝ φ´ on BΣ ´ “ BΣ `. We will argue that Σ is a punctured sphere; i.e., the boundary components of Σ co… view at source ↗
Figure 7
Figure 7. Figure 7: Depicted is the local dynamics of a conformal mating near a parabolic fixed point. easily seen to be a holomorphic double covering from Ar onto the (degenerate) annulus Bαztpu. Since Bαztpu is biholomorphic to a punctured disk, it has infinite modulus. It follows that the modulus of Ar is also infinite. Standard results for Riemann surfaces now tell us that the boundary component of Σ associated with the p… view at source ↗
Figure 8
Figure 8. Figure 8: The line complex of fpzq “ π 2 sinpzq. The graph Γ consists of three edges (colored in green, blue, and red) connecting a pair of vertices pi , pe. The Fatou components containing the singular values ˘π{2 (on the right) and the critical points (on the left) are shown in gray [PITH_FULL_IMAGE:figures/full_fig_p030_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: The modification of the line complex of fpzq “ π 2 sinpzq. After the combination (given by Theorem 4.1), we obtain the function g : CzTr Ñ C, where the dynamics in the basin U of f was replaced by the dynamics of the factor Bowen–Series map FG : DzT Ñ D coming from the modular group (see the top of Fig￾ure 10). Here φ: U Ñ D is the Riemann map used in Theorem 4.1, ψ is the David extension of the boundary c… view at source ↗
Figure 11
Figure 11. Figure 11: Remark 5.2. In this example, h has three singular values, which is one more than f. In general, however, the number of new singular values is not determined by d alone. For G P Gattrpdq one has d ` 1 “ npGqppGq: the number ppGq counts the singular boundary points of the factor tile, while npGq controls the interior ramification. Thus the group side contributes the boundary singular values coming from thes… view at source ↗
Figure 10
Figure 10. Figure 10: Top: the map g obtained after the conformal combination of the map dynamics and the group dynamics. Bottom: the domains V 8 and V 0 used, via Theo￾rem 5.1, to construct the meromorphic map h. finite number as long as #Spfq is finite. In the modular group example, ppGq “ 1, and the interior critical value is normalized to coincide with ´π{2. ↑ ↑ Do Do Do · · Do Do Do Do [PITH_FULL_IMAGE:figures/full_fig_p… view at source ↗
Figure 11
Figure 11. Figure 11: The line complex of the meromorphic map h for the function fpzq “ π 2 sinpzq P Kattrp2q (with the marked immediate basing of ´π{2) and the modular group G P Gattrp2q. The map h has a unique simple pole at 0 and three finite singular values (critical values) marked with `, ´, ▲ on the right. The corresponding critical points are marked with the respective smaller symbols on the left. The component on the l… view at source ↗
Figure 12
Figure 12. Figure 12: Illustrated is the proof of Proposition 6.8. We now construct a generating set for G0. (1) The map τ 3 : W0 Ñ W0, that sends one of the brown geodesic edges of BTp 0 to the other, is conjugated to Mi : D Ñ D under Φ´1 0 . (2) The map η preserves W1, and induces an involution of the geodesic BV 0 XW1. By the equivariance properties of the maps w ÞÑ w 4 , X, h, and by the normalization that BV 0 XWj is one … view at source ↗

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