The large scale geometry of big mapping class groups

Israel Morales Jiménez

UNAM-Oaxaca (México)

Abstract:

The mapping class group,  \[Map(S)\] , of an orientable surface  \[S\] , is the group of all isotopy classes of homeomorphisms of  \[S\] to itself. For surfaces of finite type (surfaces whose fundamental group is finitely generated),  \[Map(S)\] is a discrete topological group, it is finitely generated and its large-scale geometry has been extensively studied.

The topology of  \[Map(S)\] is more interesting if  \[S\] is an orientable infinite-type surface; it is uncountable, topologically perfect, totally disconnected, and it has the structure of a Polish group. This class of groups is called “big mapping class groups (big  \[MCGs\] )”.

Recently, based on the Rosendal framework, K. Mann and K. Rafi have started the study of the large-scale geometry of big  \[MCGs\] . In this talk I will present generalities of the theory to study the large-scale geometry of topological groups and of big  \[MCGs\] . At the end, I will present some results obtained in a joint work with Rita Jimenez Rolland where we characterize the large-scale geometry of  \[Map(S)\] when  \[S\] is a surface of infinite type with space of ends homeomorphic to  \[\omega^{\alpha}+1\] .