Classification of stable surfaces with respect to automatic continuity

Abstract

We provide a complete classification of when the homeomorphism group of a stable surface,

, has the automatic continuity property: Any homomorphism from

normal upper H normal o normal m normal e normal o left parenthesis normal upper Sigma right parenthesis H o m e o ( Σ ) $\mathrm{Homeo}({\Sigma})$

to a separable group is necessarily continuous. This result descends to a classification of when the mapping class group of

normal upper Sigma Σ $\Sigma$

has the automatic continuity property. Towards this classification, we provide a general framework for proving automatic continuity for groups of homeomorphisms. Applying this framework, we also show that the homeomorphism group of any stable, second-countable Stone space has the automatic continuity property. Under the presence of stability, this answers two questions of Mann.