Infinite Schottky groups and group actions on infinite type surfaces

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 $Γ$ 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 $Σ<em>{F}$ without planar ends can be so obtained (retrosection theorem). If $G &lt; {\rm Aut}(Σ</em>{F})$ acts freely and $Σ<em>{F}/G$ is of finite type, then we observe that it lifts to a group of automorphisms of $Ω$, for a suitable infinite Schottky uniformization of it by a infinite Schottky group $Γ$, if and only if there is a $G$-invariant collection ${\mathcal F}$ of pairwise disjoint essential simple loops on $Σ</em>{F}$ such that each connected component of $Σ_{F} \setminus {\mathcal F}$ is a finite planar surface, generalizing the situation for the case of Schottky groups of finite rank.

• The paper introduces infinite Schottky groups as countably generated free Kleinian groups associated with disjoint essential loops on infinite-type surfaces.
• It establishes equivalences between classical finite-type uniformization and the infinite-rank framework, highlighting novel criteria for lifting surface automorphisms.
• The approach bridges combinatorial topology and complex analysis, offering new insights into infinite handlebodies and the topology of hyperbolic 3-manifolds.

Infinite Schottky Groups and Group Actions on Infinite Type Surfaces

Introduction and Motivations

This paper develops a comprehensive and technically rigorous framework for Schottky groups of infinite rank—termed "infinite Schottky groups"—and systematically investigates their connection to the topology and complex analysis of infinite-type surfaces. The author presents new definitions, structural results, and equivalences generalizing the classical finite-rank Schottky uniformization to the setting of countably generated free purely loxodromic Kleinian groups.

A central theme is associating these infinite Schottky groups to collections of pairwise disjoint essential simple loops on infinite-type orientable surfaces without planar ends. The analytic and topological uniformization properties thus extend the classical retrosection theorem to a broad spectrum of surfaces, furnishing new tools for understanding group actions and automorphism lifting problems in this infinite-type context.

Schottky Systems and Loop Configurations

Central to the theory is the notion of a Schottky system of loops: a countable collection of pairwise disjoint, non-parallel essential simple loops whose removal from the surface decomposes it into planar finite-type components. These structures generalize pants decompositions and encapsulate the essential topological features required for the associated group-theoretic constructions.

For infinite-rank analogues, the paper introduces "Schottky admissible configurations" in $\widehat{\mathbb{C}}$: configurations of simple loops (often circles) that not only avoid mutual separation and accumulation, but also shrink in size—diameters tending to zero in the infinite case—ensuring a canonical limiting behavior. This combinatorics-to-analysis bridge is critical for passing from surface decompositions to discrete groups in PSL$_2(\mathbb{C})$.

Infinite Schottky Groups: Construction and Properties

An infinite Schottky group $\Gamma$ arises from a Schottky admissible configuration $\mathfrak{C} = \{C_i, C'_i\}_{i\in\mathbb{N}}$, with loxodromic generators $A_i$ acting so that $$A_i(\operatorname{Ext}(C_i)) = \operatorname{Int}(C'_i)$$. The resulting group is a purely loxodromic, countably generated free Kleinian group, acting discretely on its region of discontinuity $\Omega(\Gamma)$. Structurally, such groups have the following properties:

• There exists a $\Gamma$-invariant component $\Omega \subset \Omega(\Gamma)$, homeomorphic to a planar Riemann surface with $\Omega/\Gamma$ an infinite-type surface without planar ends.
• The complement $_2(\mathbb{C})$0 consists of topological discs with trivial $_2(\mathbb{C})$1-stabilizer.
• The limit set $_2(\mathbb{C})$2, in many cases, is totally disconnected (e.g., a Cantor set), echoing classical Schottky dynamics in the infinite setting.
• The subgroup structure reflects the infinite handlebody structure on the associated 3-manifolds.

This machinery recovers infinite-type analogues of the classical retrosection theorem: every infinite-type Riemann surface without planar ends and uniformizable by a Fuchsian group of the first kind can be conformally realized as $_2(\mathbb{C})$3 for some infinite Schottky group $_2(\mathbb{C})$4.

Lifting Automorphisms and Group Actions

A significant technical achievement is the precise criterion for when a group of surface automorphisms "lifts" through an infinite Schottky uniformization—a generalization of results well known in the finite-type case. The main theorem provides necessary and sufficient conditions (under mild hypotheses on the quotient surface):

• If $_2(\mathbb{C})$5 acts freely and the quotient $_2(\mathbb{C})$6 is of finite type, $_2(\mathbb{C})$7 lifts to a group of automorphisms of a covering domain in the Schottky uniformization if and only if there exists a $_2(\mathbb{C})$8-invariant Schottky system of loops—precisely mirroring the structure the finite-rank theory exhibits.

This result allows for a reduction of the analytic lifting problem to a combinatorial/topological one and also yields a negative answer in distinguishing infinite-type behavior from finite-type (e.g., abelian covers of surfaces of genus two can fail to have invariant Schottky systems, in contrast with finite-type cases).

Hyperbolic 3-Manifolds and Infinite Handlebodies

The correspondence between infinite Schottky groups and infinite handlebodies is clarified: given suitable additional constraints on the behavior of the defining loop configurations, the Kleinian quotient 3-manifold $_2(\mathbb{C})$9 is shown to be homeomorphic to the topological infinite handlebody naturally associated to the Schottky system on the surface.

Notably, the paper leverages deep results on planar covering surfaces and the use of "pseudocircle domains" in the characterization of the regions of discontinuity, employing tools from Kleinian group theory and low-dimensional topology.

Implications, Limitations, and Future Directions

The methodology extends uniformization, automorphism lifting, and combinatorial-to-analytic correspondences from finite-rank to infinite-rank settings. The characterization theorems and examples illuminate subtle differences between infinite and finite-type surfaces, particularly in the context of automorphism extensions and the topology of corresponding 3-manifolds.

The lifting criteria highlight obstacles unique to infinite-type surfaces, with explicit examples where anticipated properties from finite-type surface theory fail.

Future research may address:

• The development of finer invariants distinguishing among infinite Schottky uniformizations and their deformation spaces
• The exploration of non-classical configurations and their limit sets
• The study of mapping class group actions and dynamics on infinite handlebodies constructed via infinite Schottky groups
• Generalization to surfaces or complexes with more complicated end structures or other nonplanar ends

Conclusion

This work delivers a unified foundation for infinite Schottky groups and their action on infinite-type Riemann surfaces, establishing key equivalences, extension and lifting theorems, and concrete connections with infinite handlebody theory. It situates infinite Schottky groups as natural analytical and topological analogues to the well-studied finite-rank case, thus opening new directions for research in infinite-type Teichmüller theory, geometric group theory, and three-manifold topology.