Isometry groups of infinite-genus hyperbolic surfaces

TL;DR

This paper classifies isometry groups of infinite-genus hyperbolic surfaces, revealing that many such groups can be arbitrary countable groups, and explores implications for related geometric and algebraic structures.

Contribution

It provides a nearly complete classification of isometry groups for infinite-genus 2-manifolds with no planar ends, highlighting the realization of all countable groups in certain cases.

Findings

Uncountably many 2-manifolds have isometry groups that include all countable groups.

Groups of homeomorphisms, diffeomorphisms, and mapping class groups lack properties like the Tits Alternative, coherence, and residual finiteness.

Algebraic rigidity results are established for mapping class groups.

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 2-manifolds. 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.