Separability of double cosets and conjugacy classes in 3-manifold groups

Abstract

Let M=ℍ3/Γ be a hyperbolic 3-manifold of finite volume. We show that if H and K are abelian subgroups of Γ and g∈Γ, then the double coset HgK is separable in Γ. As a consequence, we prove that if M is a closed, orientable, Haken 3-manifold and the fundamental group of every hyperbolic piece of the torus decomposition of M is conjugacy separable, then so is the fundamental group of M. Invoking recent work of Agol and Wise, it follows that if M is a compact, orientable 3-manifold, then π1(M) is conjugacy separable.