Conjugacy classes of big mapping class groups

Abstract

We describe the topological behavior of the conjugacy action of the mapping class group of an orientable infinite-type surface Σ on itself by proving that:(1) all conjugacy classes of Map(Σ) are meager for every Σ,(2) Map(Σ) has a somewhere dense conjugacy class if and only if Σ has at most two maximal ends and no non-displaceable finite-type subsurfaces, (3) Map(Σ) has a dense conjugacy class if and only if Σ has a unique maximal end and no non-displaceable finite-type subsurfaces. Our techniques are based on model-theoretic methods developed by Kechris and Rosendal (Proc. Lond. Math. Soc. (3) 94 (2007) 302-350) and Truss (Proc. Lond. Math. Soc. (3) 65 (1992) 121-141).M S C 2 0 2 0 57K20, 03E15, 20E45 (primary)Let Σ be an infinite-type † surface and Map * (Σ) the group of homeomorphisms of Σ modulo isotopy. This group is called the extended (big ‡ ) mapping class group of Σ. Each Map * (Σ) acts on the curve graph 𝐶(Σ), which is the countable graph whose vertices are (isotopy classes of essential) simple closed curves in Σ and adjacency is defined by disjointness. Moreover, Map * (Σ) is isomorphic to Aut(𝐶(Σ)) [2, 6], and thus Map * (Σ) is a Polish group with respect to the permutation † A surface is of finite type if its fundamental group is finitely generated and of infinite type if not. ‡ This nomenclature is inspired by the fact that these groups are uncountable, in contrast to the countable Map(𝑆) when 𝑆 is a finite-type surface.