Conjugacy Distinguished Cosets in Hyperbolic $3$-Manifold Groups

Abstract

A subset $S$ of a group $G$ is \emph{conjugacy distinguished} if the union of all conjugates of $S$ is closed in the profinite topology on $G$. We prove that if $M = \mathbb{H}^3/Γ$ is a hyperbolic $3$-manifold of finite volume, $g \in Γ$, and $H$ is an abelian subgroup of $Γ$, then the coset $gH$ is conjugacy distinguished in $Γ$. A subset $S \subset G$ is \emph{conjugacy distinguished from a class of subgroups} if, for every $K$ in the class that is disjoint from the union of conjugates of $S$, there exists a homomorphism $φ\colon G \rightarrow F$, where $F$ is a finite group, such that $φ(K)$ is disjoint from the union of conjugates of $φ(S)$. In previous work, we proved that if $M = \mathbb{H}^3/Γ$ is a hyperbolic $3$-manifold of finite volume, then a coset of a maximal parabolic subgroup with cusp $C$ is conjugacy distinguished from the class of maximal parabolic subgroups of $Γ$ with cusps distinct from $C$. We extend this result by proving that a coset of a loxodromic subgroup is conjugacy distinguished from the class of maximal parabolic subgroups of $Γ$.