Hyperbolic 3-manifold groups are subgroup into conjugacy separable

Abstract

A group G is subgroup conjugacy distinguished (resp. subgroup into conjugacy separable) if whenever an element y (resp. a finitely generated subgroup K) is not conjugate to an element (to a subgroup) of a finitely generated subgroup H of G there exists a finite quotient G/N of G where yN (resp. KN/N) is not conjugate to an element (to a subgroup) of HN/N. We prove that the fundamental group of a hyperbolic 3-manifold is subgroup conjugacy distinguished and subgroup into conjugacy separable.