Conjugacy Distinguished Cosets in Hyperbolic $3$-Manifold Groups
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 $M = \mathbb{H}^3/\Gamma$ is a hyperbolic $3$-manifold of finite volume, $g \in \Gamma$, and $H$ is an abelian subgroup of $\Gamma$, then the coset $gH$ is conjugacy distinguished in $\Gamma$. 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 $\varphi \colon G \rightarrow F$, where $F$ is a finite group, such that $\varphi(K)$ is disjoint from the union of conjugates of $\varphi(S)$. In previous work, we proved that if $M = \mathbb{H}^3/\Gamma$ 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 $\Gamma$ 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 $\Gamma$.