Conjugacy classes of big mapping class groups.
2021
We describe the topological behavior of the conjugacy action of the mapping class group of an orientable infinite-type surface $\Sigma$ on itself. Our main results are:
(1) All conjugacy classes of $MCG(\Sigma)$ are meager for every $\Sigma$,
(2) $MCG(\Sigma)$ has a somewhere dense conjugacy class if and only if $\Sigma$ has at most two maximal ends and no non-displaceable finite-type subsurfaces,
(3) $MCG(\Sigma)$ has a dense conjugacy class if and only if $\Sigma$ has a unique maximal end and no non-displaceable finite-type subsurfaces.
Our techniques are based on model-theoretic methods developed by Kechris, Rosendal and Truss.
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