Stable commutator length on big mapping class groups

TL;DR

This paper investigates the properties of stable commutator length in infinite-type surface mapping class groups, revealing its continuity, the openness of commutator subgroups, and finite generation of abelianizations.

Contribution

It demonstrates that stable commutator length is continuous on these groups and that their commutator subgroups are open, closed, and often finitely generated.

Findings

Stable commutator length is a continuous function.

Commutator subgroups are open and closed.

Abelianizations are finitely generated in many cases.

Abstract

We study stable commutator length on mapping class groups of certain infinite-type surfaces. In particular, we show that stable commutator length defines a continuous function on the commutator subgroups of such infinite-type mapping class groups. We furthermore show that the commutator subgroups are open and closed subgroups and that the abelianizations are finitely generated in many cases. Our results apply to many popular infinite-type surfaces with locally coarsely bounded mapping class groups.