Non-amenability of mapping class groups of infinite-type surfaces and graphs

TL;DR

This paper investigates the amenability properties of various infinite-type surface and graph mapping class groups, establishing non-amenability in some cases and amenability in others, with implications for geometric group theory.

Contribution

It provides a complete characterization of non-amenability for mapping class groups of infinite-type surfaces and higher-rank graphs, and introduces examples of non-amenable stabilizers in hyperbolic groups.

Findings

Mapping class groups of infinite-type surfaces are non-amenable.

Certain mapping class groups of trees or rank-one graphs are amenable.

An example of a non-amenable stabilizer of a point at infinity in a hyperbolic Polish group.

Abstract

This paper completely determines the non-amenability of the mapping class groups of infinite-type surfaces, the mapping class groups of locally finite infinite graphs of higher ranks, gives an example of non-amenable stabiliser of a point at infinity of a coarsely bounded generated hyperbolic Polish group, and exhibits a class of mapping class groups of trees or rank-one graphs that are amenable.