Conjugacy classes of big mapping class groups

TL;DR

This paper investigates the conjugacy classes of big mapping class groups of infinite-type surfaces, revealing conditions for the existence of dense and meager conjugacy classes using model-theoretic methods.

Contribution

It characterizes when the mapping class group has dense or meager conjugacy classes based on the surface’s topological features, a novel application of model theory.

Findings

All conjugacy classes are meager for every surface.

Existence of a somewhere dense conjugacy class depends on the surface's ends.

A dense conjugacy class exists if the surface has a unique maximal end.

Abstract

We describe the topological behavior of the conjugacy action of the mapping class group of an orientable infinite-type surface $\Sigma$ on itself. Our main results are: (1) All conjugacy classes of $MCG(\Sigma)$ are meager for every $\Sigma$, (2) $MCG(\Sigma)$ has a somewhere dense conjugacy class if and only if $\Sigma$ has at most two maximal ends and no non-displaceable finite-type subsurfaces, (3) $MCG(\Sigma)$ has a dense conjugacy class if and only if $\Sigma$ has a unique maximal end and no non-displaceable finite-type subsurfaces. Our techniques are based on model-theoretic methods developed by Kechris, Rosendal and Truss.