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

TL;DR

This paper proves that certain double cosets in hyperbolic 3-manifold groups are separable and extends conjugacy separability to fundamental groups of all compact, orientable 3-manifolds, building on recent breakthroughs.

Contribution

It establishes the separability of double cosets in hyperbolic 3-manifold groups and proves conjugacy separability for fundamental groups of all compact, orientable 3-manifolds.

Findings

Double cosets HgK are separable in hyperbolic 3-manifold groups.

Fundamental groups of hyperbolic pieces are conjugacy separable.

All compact, orientable 3-manifold groups are conjugacy separable.

Abstract

Let M = H^3 / \Gamma be a hyperbolic 3-manifold of finite volume. We show that if H and K are abelian subgroups of \Gamma and g is in \Gamma, then the double coset HgK is separable in \Gamma. 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 \pi_1(M) is conjugacy separable.