Big mapping class groups are not extremely amenable

TL;DR

This paper demonstrates that large mapping class groups are generally not extremely amenable, except in trivial cases, using the Kechris-Pestov-Todorčević framework, highlighting limitations in their dynamical properties.

Contribution

It applies the Kechris-Pestov-Todorčević machinery to establish non-extreme amenability of big and pure mapping class groups for surfaces with genus at least one.

Findings

Big mapping class groups are not extremely amenable unless trivial.

Pure and compactly supported mapping class groups are not extremely amenable for genus ≥ 1.

The results specify conditions under which mapping class groups lack extreme amenability.

Abstract

This paper uses the renowned Kechris-Pestov-Todor\v{c}evi\'{c} machinery to show that (big) mapping class groups are not extremely amenable unless the underlying surface is a sphere or a once-punctured sphere, or equivalently when the mapping class group is trivial. The same techniques also show that the pure mapping class groups, as well as compactly supported mapping class groups, of a surface with genus at least one can never be extremely amenable.