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Classification of stable surfaces with respect to automatic continuity
- Part of
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- Journal:
- Compositio Mathematica / Volume 162 / Issue 3 / March 2026
- Published online by Cambridge University Press:
- 23 June 2026, pp. 669-702
- Print publication:
- March 2026
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- Article
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We provide a complete classification of when the homeomorphism group of a stable surface,
$\Sigma$, has the automatic continuity property: Any homomorphism from 
$\mathrm{Homeo}({\Sigma})$ to a separable group is necessarily continuous. This result descends to a classification of when the mapping class group of 
$\Sigma$ 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.
