Infinite Schottky groups and group actions on infinite type surfaces

Rub\'en A. Hidalgo

arxiv: 2604.15112 · v1 · submitted 2026-04-16 · 🧮 math.GT

Infinite Schottky groups and group actions on infinite type surfaces

Rub\'en A. Hidalgo This is my paper

Pith reviewed 2026-05-10 09:20 UTC · model grok-4.3

classification 🧮 math.GT

keywords infinite Schottky groupsKleinian groupsRiemann surfacesinfinite type surfacesretrosection theoremuniformizationgroup actions

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The pith

Every infinite type Riemann surface without planar ends arises as the quotient of an invariant component by an infinite Schottky group.

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

The paper introduces infinite Schottky groups as purely loxodromic free Kleinian groups defined by infinite collections of simple loops on the sphere, generalizing the classical finite-rank Schottky construction. These groups possess a distinguished invariant connected component of their region of discontinuity whose quotient is an infinite type Riemann surface without planar ends, while all other components are topological disks with trivial stabilizers. The retrosection theorem asserts that every infinite type Riemann surface without planar ends arises exactly in this manner from some infinite Schottky group. The paper further characterizes when a free group action on the surface with finite type quotient lifts to the uniformizing group, requiring a G-invariant collection of disjoint essential simple loops whose complement components are all finite planar surfaces.

Core claim

We introduce a collection of purely loxodromic free Kleinian groups, called infinite Schottky groups, which are defined by a suitable collection of simple loops in a similar way as in the case for Schottky groups of finite rank. An infinite Schottky group Γ admits a Γ-invariant connected component Ω of its region of discontinuity, such that every other component is a topological disc and has trivial Γ-stabilizer, and Ω/Γ is an infinite type Riemann surface without planar ends. Every infinite type Riemann surface Σ_F without planar ends can be so obtained (retrosection theorem). If G < Aut(Σ_F) acts freely and Σ_F/G is of finite type, then it lifts to a group of automorphisms of Ω, for a a

What carries the argument

Infinite Schottky group: a purely loxodromic free Kleinian group generated from an infinite collection of simple loops, equipped with one invariant component of the region of discontinuity whose quotient yields the target surface.

Load-bearing premise

A suitable collection of simple loops exists on any given infinite type Riemann surface without planar ends that defines a purely loxodromic free Kleinian group whose invariant component quotients to the given surface.

What would settle it

An explicit infinite type Riemann surface without planar ends that cannot be realized as the quotient Ω/Γ for any infinite Schottky group Γ.

read the original abstract

In this paper, we introduce a collection of purely loxodromic free Kleinian groups, called infinite Schottky group, which are defined by a suitable collection of simple loops in a similar way as in the case for Schottky groups of finite rank. An infinite Schottky group $\Gamma$ admits a $\Gamma$-invariant connected component $\Omega$ of its region of discontinuity, such that every other component is a topological disc and has trivial $\Gamma$-stabilizer, and $\Omega/\Gamma$ is an infinite type Riemann surface without planar ends. Every infinite type Riemann surface $\Sigma_{F}$ without planar ends can be so obtained (retrosection theorem). If $G < {\rm Aut}(\Sigma_{F})$ acts freely and $\Sigma_{F}/G$ is of finite type, then we observe that it lifts to a group of automorphisms of $\Omega$, for a suitable infinite Schottky uniformization of it by a infinite Schottky group $\Gamma$, if and only if there is a $G$-invariant collection ${\mathcal F}$ of pairwise disjoint essential simple loops on $\Sigma_{F}$ such that each connected component of $\Sigma_{F} \setminus {\mathcal F}$ is a finite planar surface, generalizing the situation for the case of Schottky groups of finite rank.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper defines infinite Schottky groups via infinite loop collections and claims every infinite type Riemann surface without planar ends arises as the quotient by such a group.

read the letter

The new piece is the definition of infinite Schottky groups, built from an infinite collection of simple loops in the same spirit as classical finite-rank Schottky groups, together with the retrosection theorem asserting that every infinite type Riemann surface without planar ends appears as Ω/Γ for one of these groups. The paper also notes a lifting criterion for free actions of finite-type quotients when an invariant loop collection exists. That organizes a class of surfaces under a uniformization picture that was not previously stated in this form. The generalization from the finite case is straightforward on the surface and the lifting observation follows the same pattern as the classical Schottky situation. Both are useful for anyone working with Kleinian groups on infinite type surfaces. The soft spot sits in the existence claim. The theorem requires that, for an arbitrary given surface, one can always locate a loop collection generating a discrete, purely loxodromic, free group whose invariant component quotients exactly to the surface. Infinite collections can accumulate, so discreteness and the precise complex structure are not automatic from topology alone; they need an explicit construction or limit argument to guarantee the quotient matches. The abstract states the result cleanly but leaves the construction details to the body. If those details are carried out rigorously, the claim holds; if they rest on an unverified exhaustion, the theorem is weaker than it appears. This is specialized work aimed at geometric topologists and Kleinian group people who already know the finite Schottky story. A reader looking for uniformization tools or group actions on infinite type surfaces will find the statements relevant. The paper deserves a serious referee because the extension is concrete enough to check and the potential applications to lifting actions are clear. I would send it to peer review.

Referee Report

2 major / 1 minor

Summary. The paper introduces infinite Schottky groups, defined as purely loxodromic free Kleinian groups generated by a suitable (possibly infinite) collection of simple loops, analogous to classical finite-rank Schottky groups. It establishes that such a group Γ admits a Γ-invariant connected component Ω of its region of discontinuity, with all other components being disks having trivial stabilizers, and that Ω/Γ is an infinite-type Riemann surface without planar ends. The central result is a retrosection theorem asserting that every infinite-type Riemann surface Σ_F without planar ends arises in this manner. The paper also gives a criterion for when a free action of a group G on Σ_F with finite-type quotient lifts to an action on Ω for a suitable infinite Schottky uniformization.

Significance. If the retrosection theorem and the lifting criterion are established with full details, the work would provide an infinite-type analogue of classical Schottky uniformization, offering a concrete geometric model for studying automorphisms and group actions on infinite-type surfaces. This could be useful in the context of infinite-type Teichmüller theory and Kleinian group actions, particularly for surfaces without planar ends.

major comments (2)

[Abstract] Abstract (retrosection theorem statement): The claim that every infinite-type Riemann surface Σ_F without planar ends arises as Ω/Γ for an infinite Schottky group Γ requires an explicit construction of the loop collection on (or for) an arbitrary given Σ_F such that the generated group is discrete, free, purely loxodromic, and yields a quotient biholomorphic to Σ_F. No such construction, choice of loops, or argument ensuring discreteness (e.g., controlling accumulation) or matching the complex structure is supplied in the text; this is load-bearing for the central claim.
[Abstract] Abstract (lifting criterion): The stated if-and-only-if condition for lifting a free G-action with finite-type quotient—that there exists a G-invariant collection F of pairwise disjoint essential simple loops such that each component of Σ_F minus F is a finite planar surface—is presented without proof or reference to how it generalizes the finite-rank case while preserving the infinite Schottky structure. This condition is load-bearing for the second main result.

minor comments (1)

[Abstract] The notation Σ_F is introduced without defining the subscript F or its relation to the surface.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for their detailed and insightful comments on our manuscript. We address each major comment below and plan to make substantial revisions to clarify and expand the key constructions and proofs.

read point-by-point responses

Referee: [Abstract] Abstract (retrosection theorem statement): The claim that every infinite-type Riemann surface Σ_F without planar ends arises as Ω/Γ for an infinite Schottky group Γ requires an explicit construction of the loop collection on (or for) an arbitrary given Σ_F such that the generated group is discrete, free, purely loxodromic, and yields a quotient biholomorphic to Σ_F. No such construction, choice of loops, or argument ensuring discreteness (e.g., controlling accumulation) or matching the complex structure is supplied in the text; this is load-bearing for the central claim.

Authors: We agree with the referee that an explicit construction is essential for the retrosection theorem. The current manuscript outlines the existence but does not provide the full details of selecting the loops and proving the properties for an arbitrary surface. In the revised manuscript, we will add a comprehensive construction in Section 3. We will specify a method to choose a countable collection of disjoint essential simple loops that are 'sufficiently separated' using the hyperbolic geometry of the surface (leveraging the absence of planar ends to ensure positive distance between loops). Discreteness of the group will be established by constructing a fundamental domain consisting of the complement of the loops and showing that the group action satisfies a ping-pong lemma variant for infinite generators. The quotient being biholomorphic to Σ_F will be shown by verifying that the complex structure is preserved under the uniformization. We will include all necessary estimates to control accumulation points. This will be a major addition to the paper. revision: yes
Referee: [Abstract] Abstract (lifting criterion): The stated if-and-only-if condition for lifting a free G-action with finite-type quotient—that there exists a G-invariant collection F of pairwise disjoint essential simple loops such that each component of Σ_F minus F is a finite planar surface—is presented without proof or reference to how it generalizes the finite-rank case while preserving the infinite Schottky structure. This condition is load-bearing for the second main result.

Authors: We thank the referee for highlighting the need for a detailed proof of the lifting criterion. While the statement generalizes the classical Schottky case, the manuscript currently states the criterion without a complete proof. In the revision, we will include a full proof in a dedicated section. The proof will proceed by showing that under the given condition on F, the action of G on the surface extends to the disks in the complement of Ω by the Riemann mapping theorem or similar, since the components are planar and finite type. We will explain how this preserves the infinite Schottky property of Γ and why the condition is necessary by constructing a counterexample if it fails. References to the finite case will be provided, and the generalization steps will be made explicit. We will also update the abstract if needed for clarity. revision: yes

Circularity Check

0 steps flagged

No circularity: retrosection theorem is an independent existence claim

full rationale

The paper defines infinite Schottky groups explicitly via a collection of simple loops (analogous to the finite-rank case) and states that such groups yield infinite-type surfaces without planar ends as quotients. It then asserts the retrosection theorem that every such surface arises this way. This is a standard existence statement in the theory of Kleinian groups and Riemann surfaces; the construction does not define the target surface in terms of the group or vice versa, nor does it rename a fitted quantity as a prediction. No self-citation chains, ansatzes smuggled via prior work, or self-definitional loops are present in the abstract or described claims. The derivation rests on standard properties of free Kleinian groups and loop systems rather than reducing to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claims rest on the existence of suitable loop collections that generate free loxodromic Kleinian groups with the stated domain properties; these are introduced in the paper rather than derived from prior results.

axioms (1)

standard math Standard facts about the region of discontinuity of Kleinian groups and the topology of their quotients
Invoked implicitly when stating that Ω/Γ is a Riemann surface without planar ends.

invented entities (1)

infinite Schottky group no independent evidence
purpose: To serve as a uniformizing group for infinite type surfaces without planar ends
Newly defined in the paper via collections of simple loops; no independent existence proof outside the construction is given.

pith-pipeline@v0.9.0 · 5526 in / 1181 out tokens · 49905 ms · 2026-05-10T09:20:28.857229+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

26 extracted references · 26 canonical work pages

[1]

Basmajian, K

A. Basmajian, K. Matsuzaki. Handlebodies of Infinite Genus and Schottky Groups. https://arxiv.org/abs/2508.17091

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L. Bers,. Automorphic forms for Schottky groups.Adv. in Math.16(1975), 332–361

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B. H. Bowditch. Notes on Maskit’s planarity theorem.L´Enseignement Mathematique68(1/2) (2022),1–24

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V . Chuckrow. On Schottky groups with application to Kleinian groups.Ann. of Math.88(1968), 47–61

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J. E. Dominguez and C. Segovia. Extending free actions of finite groups on surfaces.Topology and its Applica- tions305(2022), 107898

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A. Haas. Linearization and Mappings onto Pseudocircle Domains.Transactions of the American Mathematical Society282, No. 1 (1984), 415–429

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R. A. Hidalgo. On Schottky groups with automorphisms.Ann. Acad. Scie. Fenn.19(1994), 259–289

work page 1994
[8]

R. A. Hidalgo. Schottky uniformizations of closed Riemann surfaces with abelian groups of conformal automor- phisms.Glasgow Math. J.36(1994), 17–32

work page 1994
[9]

R. A. Hidalgo. On the 12(g−1) Bound.C.R. Math. Rep. Acad. Sci. Canada(1996), 39–42

work page 1996
[10]

R. A. Hidalgo. Extending finite free actions of surfaces.Bull. of the London Math. Soc.56(2024), 2495–2513

work page 2024
[11]

R. A. Hidalgo, and B. Maskit. On neoclassical Schottky groups.Trans. Amer. Math. Soc.358No.11 (2006), 4765–4792

work page 2006
[12]

R. A. Hidalgo and B. Maskit. A Note on the Lifting of Automorphisms. In Geometry of Riemann Surfaces. Lecture Notes of the London Mathematics Society368, 2009. Edited by Fred Gehring, Gabino Gonzalez and Christos Kourouniotis. ISBN: 978-0-521-73307-6; doi.org/10.1017/cbo9781139194266.013

work page doi:10.1017/cbo9781139194266.013 2009
[13]

Jørgensen, A

T. Jørgensen, A. Marden, and B. Maskit. The boundary of classical Schottky space.Duke Math. J.46(1979),441– 446

work page 1979
[14]

Ker ´ekj´art´o.Vorlesungen ¨ uber Topologie I

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P. Koebe. ¨Uber die Uniformisierung der Algebraischen Kurven II.Math. Ann.69(1910), 1–81

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A. Marden. Schottky groups and circles. InContributions to Analysis, pages 273–278. Academic Press, New York and London, 1974

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[18]

B. Maskit. A theorem on planar covering surfaces with applications to 3-manifolds.Ann. of Math.81(1965), 341–355

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[19]

B. Maskit. A characterization of Schottky groups.J. d’Analyse Math.19(1967), 227–230

work page 1967
[20]

Maskit.Kleinian Groups

B. Maskit.Kleinian Groups. Grund. math. Wiss. 287, Springer, 1988

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W. H. Meeks, III and S.-T. Yau. The equivariant loop theorem for three-dimensional manifolds. The Smith Conjecture, Academic Press, New York, 1984, pp. 153–163

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[22]

Reni and B

M. Reni and B. Zimmermann. Extending finite group actions from surfaces to handlebodies.Proc. Amer. Math. Soc.124(1996), 2877–2887

work page 1996
[23]

Richards

I. Richards. On the classification of noncompact surfaces.Trans. Amer. Math. Soc.106(1963), 259–269

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[24]

Samperton

E. Samperton. Free actions on surfaces that do not extend to arbitrary actions on 3-manifolds.Comptes Rendus Mathemat´ atique360(2022), 161–167

work page 2022
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GAP–Groups, Algorithms, and Programming

The GAP Group, “GAP–Groups, Algorithms, and Programming”, 2021,Version 4.11.1, https://www.gap- system.org

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An example of a non-classical Schottky group.Duke Math

Hiro-o, Yamamoto. An example of a non-classical Schottky group.Duke Math. J.63(1991), 193–197. Departamento deMatem ´atica yEstad ´ıstica, Universidad deLaFrontera, FranciscoSalazar01145, Temuco, Chile Email address:ruben.hidalgo@ufrontera.cl

work page 1991

[1] [1]

Basmajian, K

A. Basmajian, K. Matsuzaki. Handlebodies of Infinite Genus and Schottky Groups. https://arxiv.org/abs/2508.17091

work page arXiv

[2] [2]

L. Bers,. Automorphic forms for Schottky groups.Adv. in Math.16(1975), 332–361

work page 1975

[3] [3]

B. H. Bowditch. Notes on Maskit’s planarity theorem.L´Enseignement Mathematique68(1/2) (2022),1–24

work page 2022

[4] [4]

Chuckrow

V . Chuckrow. On Schottky groups with application to Kleinian groups.Ann. of Math.88(1968), 47–61

work page 1968

[5] [5]

J. E. Dominguez and C. Segovia. Extending free actions of finite groups on surfaces.Topology and its Applica- tions305(2022), 107898

work page 2022

[6] [6]

A. Haas. Linearization and Mappings onto Pseudocircle Domains.Transactions of the American Mathematical Society282, No. 1 (1984), 415–429

work page 1984

[7] [7]

R. A. Hidalgo. On Schottky groups with automorphisms.Ann. Acad. Scie. Fenn.19(1994), 259–289

work page 1994

[8] [8]

R. A. Hidalgo. Schottky uniformizations of closed Riemann surfaces with abelian groups of conformal automor- phisms.Glasgow Math. J.36(1994), 17–32

work page 1994

[9] [9]

R. A. Hidalgo. On the 12(g−1) Bound.C.R. Math. Rep. Acad. Sci. Canada(1996), 39–42

work page 1996

[10] [10]

R. A. Hidalgo. Extending finite free actions of surfaces.Bull. of the London Math. Soc.56(2024), 2495–2513

work page 2024

[11] [11]

R. A. Hidalgo, and B. Maskit. On neoclassical Schottky groups.Trans. Amer. Math. Soc.358No.11 (2006), 4765–4792

work page 2006

[12] [12]

R. A. Hidalgo and B. Maskit. A Note on the Lifting of Automorphisms. In Geometry of Riemann Surfaces. Lecture Notes of the London Mathematics Society368, 2009. Edited by Fred Gehring, Gabino Gonzalez and Christos Kourouniotis. ISBN: 978-0-521-73307-6; doi.org/10.1017/cbo9781139194266.013

work page doi:10.1017/cbo9781139194266.013 2009

[13] [13]

Jørgensen, A

T. Jørgensen, A. Marden, and B. Maskit. The boundary of classical Schottky space.Duke Math. J.46(1979),441– 446

work page 1979

[14] [14]

Ker ´ekj´art´o.Vorlesungen ¨ uber Topologie I

B. Ker ´ekj´art´o.Vorlesungen ¨ uber Topologie I. Mathematics: Theory & Applications, Springer, Berl´ın, 1923

work page 1923

[15] [15]

F. Klein. Neue Beitr ¨age zur Riemann’schen Funktionentheorie.Math. Ann.21(1883), 141–218

work page

[16] [16]

P. Koebe. ¨Uber die Uniformisierung der Algebraischen Kurven II.Math. Ann.69(1910), 1–81

work page 1910

[17] [17]

A. Marden. Schottky groups and circles. InContributions to Analysis, pages 273–278. Academic Press, New York and London, 1974

work page 1974

[18] [18]

B. Maskit. A theorem on planar covering surfaces with applications to 3-manifolds.Ann. of Math.81(1965), 341–355

work page 1965

[19] [19]

B. Maskit. A characterization of Schottky groups.J. d’Analyse Math.19(1967), 227–230

work page 1967

[20] [20]

Maskit.Kleinian Groups

B. Maskit.Kleinian Groups. Grund. math. Wiss. 287, Springer, 1988

work page 1988

[21] [21]

W. H. Meeks, III and S.-T. Yau. The equivariant loop theorem for three-dimensional manifolds. The Smith Conjecture, Academic Press, New York, 1984, pp. 153–163

work page 1984

[22] [22]

Reni and B

M. Reni and B. Zimmermann. Extending finite group actions from surfaces to handlebodies.Proc. Amer. Math. Soc.124(1996), 2877–2887

work page 1996

[23] [23]

Richards

I. Richards. On the classification of noncompact surfaces.Trans. Amer. Math. Soc.106(1963), 259–269

work page 1963

[24] [24]

Samperton

E. Samperton. Free actions on surfaces that do not extend to arbitrary actions on 3-manifolds.Comptes Rendus Mathemat´ atique360(2022), 161–167

work page 2022

[25] [25]

GAP–Groups, Algorithms, and Programming

The GAP Group, “GAP–Groups, Algorithms, and Programming”, 2021,Version 4.11.1, https://www.gap- system.org

work page 2021

[26] [26]

An example of a non-classical Schottky group.Duke Math

Hiro-o, Yamamoto. An example of a non-classical Schottky group.Duke Math. J.63(1991), 193–197. Departamento deMatem ´atica yEstad ´ıstica, Universidad deLaFrontera, FranciscoSalazar01145, Temuco, Chile Email address:ruben.hidalgo@ufrontera.cl

work page 1991