Convex cocompact subgroups of the Goeritz group

This talk is about hyperbolicity of surface group extensions and a question of Farb-Mosher about whether purely pseudo-Anosov subgroups of mapping class groups are convex cocompact. We’ll explain this problem and give an answer for subgroups of the genus-2 Goeritz group, which is the group of mapping classes of a genus-2 surface that extend to the genus-2 Heegaard splitting of the 3-sphere. In addition we give a refined version of the Nielsen-Thurston classification for the Goeritz group.