Hyperbolic 3-Manifolds Groups are Subgroup Conjugacy Separable

Abstract

A group G is called subgroup conjugacy separable if for every pair of non-conjugate finitely generated subgroups of G, there exists a finite quotient of G where the images of these subgroups are not conjugate. It is proved that the fundamental group of a hyperbolic 3-manifold (closed or with cusps) is subgroup conjugacy separable.

Infinitely many virtual geometric triangulations

“…Given a peripheral subgroup $$K$$ and a single element $$g \in \Gamma$$ that is disjoint from every conjugate of $$K$$, it is straightforward to find a finite quotient that witnesses this disjointness [18, Lemma 4.5]. Given non‐conjugate subgroups $$H$$ and $$K$$, Chagas and Zalesskii find a finite quotient of $$\Gamma$$ where their images are not conjugate [4]. Given a pair of non‐conjugate peripheral subgroups $$H$$ and $$K$$, Wilton and Zalesskii use an argument of Hamilton to construct a finite quotient $$\varphi \colon \Gamma \rightarrow G$$, such that non‐trivial elements of $$\varphi (H)$$ and $$\varphi (K)$$ always lie in distinct conjugacy classes [31, Lemma 4.6].…”