Conjugacy classes of big mapping class groups

We describe the topological behavior of the conjugacy action of the mapping class group of an orientable infinite-type surface Σ Σ on itself by proving that: (1) all conjugacy classes of Map ( Σ ) Map(Σ ) are meager for every Σ Σ , (2) Map ( Σ ) Map(Σ ) has a somewhere dense conjugacy class if and only if Σ Σ has at most two maximal ends and no non-displaceable finite-type subsurfaces, (3) Map ( Σ ) Map(Σ ) has a dense conjugacy class if and only if Σ 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).