Handlebodies of Infinite Genus and Schottky Groups

TL;DR

This paper explores the topology and uniformization of infinite genus surfaces and handlebodies, establishing conditions under which they can be uniformized by infinitely generated Schottky groups and discussing uniqueness and limitations.

Contribution

It introduces definitions for infinite genus handlebodies and Schottky groups, and characterizes when such surfaces admit uniformizations by classical Schottky groups, including conditions for uniqueness.

Findings

Any infinite genus surface with non-planar ends can be uniformized by an infinitely generated Schottky group.

Surfaces with bounded pants decompositions admit quasiconformal uniformizations with unique conformal classes.

Some infinite genus surfaces without bounded pants decompositions do not admit Schottky uniformizations.

Abstract

In this paper, we begin an investigation of infinite genus handlebodies, infinitely generated Schottky groups, and related uniformization questions by giving appropriate definitions for them. There are uncountably many topological types of infinite genus surfaces with non-planar ends. We show that any such surface and any infinite genus handlebody can be topologically uniformized by an infinitely generated classical Schottky group. We next show that an infinite genus Riemann surface with non-planar ends admitting a bounded pants decomposition can be quasiconformally uniformized by a classical Schottky group. In addition, the conformal equivalence class of such a uniformization is unique.If the assumption of bounded pants decomposition is removed we supply examples of such Riemann surfaces that do not admit a quasiconformal uniformization by a Schottky group.