Isometry groups of infinite-genus hyperbolic surfaces

Abstract

Given a 2-manifold, a fundamental question to ask is which groups can be realized as the isometry group of a Riemannan metric of constant curvature on the manifold. In this paper, we give a nearly complete classification of such groups for infinite-genus 2-manifolds with no planar ends. Surprisingly, we show there is an uncountable class of such 2-manifolds where every countable group can be realized as an isometry group (namely, those with self-similar end spaces). We apply this result to obtain obstructions to standard group theoretic properties for the groups of homeomorphisms, diffeomorphisms, and the mapping class groups of such 2manifolds. For example, none of these groups satisfy the Tits Alternative; are coherent; are linear; are cyclically or linearly orderable; or are residually finite. As a second application, we give an algebraic rigidity result for mapping class groups.

A note on subgroups of the Loch Ness Monster Surface's Mapping Class Group

“…The following theorem follows from a deep result of Aougab-Patel-Vlamis [APV21] about hyperbolic isometry groups of infinite type surfaces:…”

Isospectral hyperbolic surfaces of infinite genus

“…To construct the covering surface we will rely on the construction of infinite-genus hyperbolic surfaces with a given (finite) isometry group due to Aougab, Patel and Vlamis [APV20].…”

“…We will use the same construction also for the proof of Theorem B. The main difference with Theorem A is that these surfaces admit hyperbolic structures with a countably infinite isometry group, as shown by Aougab, Patel and Vlamis [APV20]. While this will allow us to construct infinite isospectral families, it will also mean that we won't be able to apply Sunada's result directly.…”

Most big mapping class groups fail the Tits Alternative

“…5.30]. Further examples come from the work of Aougab, Patel and Vlamis [2], who constructed surfaces X such that every countable group embeds in Map(X).…”