Mark Hagen

Separability (and mapping class groups)

I'll review the notions of residual finiteness and separability and discuss some examples (things like free groups and surface groups). I'll discuss the relationship between separability of quasiconvex subgroups of hyperbolic groups and Gromov's question about residual finiteness of hyperbolic groups, in particular a result of Agol-Groves-Manning which is in principle evidence against a positive answer to the question. Then we turn to mapping class groups of surfaces, and the question of Reid about separability of convex-cocompact subgroups. I will discuss a result (joint work with J Behrstock, A Martin, and A Sisto) that *arguably* relates Reid's question to Gromov's question, somewhat analogously to the Agol-Groves-Manning result. I will finally discuss some examples of separable convex-cocompact subgroups of mapping class groups (constructed in joint work with A Sisto). If there's time to prove anything, it will be a criterion for detecting separability in outer automorphism groups, which is a small generalisation of the criterion concocted by Grossman in the 1970s to prove that mapping class groups are residually finite.