Around big mapping class groups

Mapping class groups associated to surfaces whose fundamental group is not finitely generated are called "big". These arise for example when studying groups of homeomorphisms of the plane with a bounded orbit. In this talk we discuss some of the basic properties of big MCS,  the simplices on which these groups act and how this simplicial actions can be used to prove, for example, that the space of non-trivial quasimorphism of a big mapping class group is infinite dimentional. Since this is a relatively recent subject, if time permits we will discuss some of the main open questions.