Automatic continuity of pure mapping class groups

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.