| Given a surface \(S\), the mapping class group of \(S\) is the group of isotopy classes of self-homeomorphisms of \(S\). If \(S\) of finite type this group has been very well studied by geometric group theorists and low-dimensional topologists. If \(S\) if of infinite type, it has only been recently that the (big) mapping class group has been studied, both from the topological group theoretic side and from the large-scale geometric side. In this talk we introduce some of the techniques used on finite-type surfaces and how well (or not) they have been adapted to infinite-type surfaces. In particular, we talk about the countable structures on which the big mapping class group acts and how to start studying its large-scale geometry. |
