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 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.

arXiv comment49 pages, 5 figures. v2: incorporates referee's comments, final version, accepted for publication

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Cited by in corpus (5)

• Big mapping class groups: an overview
• Nielsen realization for infinite-type surfaces
• Mapping class groups with the Rokhlin property
• Most big mapping class groups fail the Tits Alternative
• A new construction of subgroups of big mapping class groups