Separable subgroups of mapping class groups 1 Separable subgroups of mapping class groups

We investigate separability questions for the mapping class group of a sur-face. While this group is not subgroup separable in general, we prove a large family of interesting subgroups are separable. This includes many classically studied subgroups such as solvable subgroups, Heegaard and Handlebody groups, geometric subgroups, and all the terms in the Johnson filtration. 1 Introduction and main results A subgroup H of G is said to be separable in G if it can be expressed as an in-tersection of finite index subgroups of G. If the trivial subgroup {1} is separable, G is residually finite. More generally, if every finitely generated subgroup of G is separable, G is subgroup separable (or LERF)