Topological behaviour of conjugacy classes of big mapping class groups

Classical mapping class groups, i.e. for surfaces of finite type, are well-studied objects: they are discrete groups expressing the symmetries of the surface. When we turn our attention to surfaces of infinite type, the situation changes drastically: In particular, the mapping class groups are now "big" in the sense that they are uncountable, not finitely generated, and not even compactly generated.

This means that we loose some of the tools that are used in the classical case but also that we can ask many new questions, for example such coming from the theory of Polish groups: Are big mapping class groups automatically continuous? Amenable? When considering the conjugacy action of a big mapping class group on itself, can there be comeager, dense, or at least somewhere dense orbits?

In this talk, I will give a short introduction to surfaces of infinite type and big mapping class groups and then answer some of the new questions, based on joint work with Jesús Hernández Hernández, Michael Hrušák, Israel Morales, Manuel Sedano, and Ferrán Valdez.