the mapping class group, $\mathrm{map}(s)$, of a surface $s$, is the group of all isotopy classes of homeomorphisms of $s$ to itself. a mapping class group is a topological group with the quotient topology inherited from the quotient map of $\mathrm{homeo}(s)$ with the compact-open topology.

for surfaces of finite type, $\mathrm{map}(s)$ is countable and discrete. surprisingly, the topology of $\mathrm{map}(s)$ is more interesting if $s$ is an infinite-type surface; it is uncountable, topologically perfect, totally disconnected, and more importantly, has the structure of a polish group. in recent literature, this last class of groups is called "big mapping class groups.''

in this talk, i will give a brief introduction to big mapping class groups and explain our results on the topological structure of conjugacy classes. this was a joint work with jesús hernández hernández, michael hrušák, manuel sedano, and ferrán valdez.