Automatic continuity of pure mapping class groups (Q6574912)

For a metrisable, connected, orientable surface \(S\), let \(\mathrm{Map}(S)\) be the mapping class group of \(S\), i.e. the group of homeomorphisms of \(S\) fixing the boundary pointwise modulo isotopy relative to the boundary, and \(\mathrm{PMap}(S)\) the pure mapping class group of \(S\), i.e. the subgroup of \(\mathrm{Map}(S)\) fixing the ends. If \(S\) is a connected sum of finitely many discs with handles with a finite type surface and has finitely many ends accumulated by genus then \(\mathrm{PMap}(S)\) and \(\mathrm{Map}(S)\) have automatic continuity, with the converse also holding for the former group. A corresponding characterisation is given for the closure of the subgroup of compactly supported classes. \(\mathrm{Map}(S)\) does not have automatic continuity if \(S\) is of infinite type with finitely many ends and no noncompact boundary components.