Automatic continuity of pure mapping class groups

TL;DR

This paper classifies infinite-type surfaces based on whether their pure mapping class groups exhibit automatic continuity, revealing precise conditions and extending understanding of topological group properties in geometric topology.

Contribution

It provides a complete classification of surfaces for which the pure mapping class group and its subgroups have automatic continuity, including cases with noncompact boundary.

Findings

Pure mapping class groups have automatic continuity for certain infinite-type surfaces.

Mapping class groups lack automatic continuity for surfaces with finitely many ends and no noncompact boundary.

Homomorphisms from certain subgroups to countable groups are trivial in some cases.

Abstract

We completely classify the orientable infinite-type surfaces $S$ such that $\operatorname{PMap}(S)$, the pure mapping class group, has automatic continuity. This classification includes surfaces with noncompact boundary. In the case of surfaces with finitely many ends and no noncompact boundary components, we prove the mapping class group $\operatorname{Map}(S)$ does not have automatic continuity. We also completely classify the surfaces such that $\overline{\operatorname{PMap}_c(S)}$, the subgroup of the pure mapping class group composed of elements with representatives that can be approximated by compactly supported homeomorphisms, has automatic continuity. In some cases when $\overline{\operatorname{PMap}_c(S)}$ has automatic continuity, we show any homomorphism from $\overline{\operatorname{PMap}_c(S)}$ to a countable group is trivial.