Séance Séminaire

Séminaire Gaston Darboux

Subgroups of the mapping class group

This talk is mainly about the surface subgroups. The geometry of surface subgroups of the mapping class group is essentially equivalent to the geometry of surface-by-surface groups. There are many open questions, for example it is not known if a surface-by-surface group can be hyperbolic, or indeed if it can be ``atoroidal'' in the sense of not containing any free abelian subgroup of rank 2. However one can show that, for fixed genera, there are only finitely many isomorphism classes of atoroidal surface-by-surface groups. The proof uses the geometry of the curve graph of Harvey and ideas of Masur and Minsky et al. from the proof of the ending lamination conjecture.