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 $M={\mathbb{H}}^3/\hat{I}“$ is a hyperbolic $3$-manifold of finite volume, $g\in \hat{I}“$, and $H$ is an abelian subgroup of $\hat{I}“$, then the coset $gH$ is conjugacy distinguished in $\hat{I}“$. 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 $\ddot{I}†:G\to F$, where $F$ is a finite group, such that $\ddot{I}†(K)$ is disjoint from the union of conjugates of $\ddot{I}†(S)$. In previous work, we proved that if $M={\mathbb{H}}^3/\hat{I}“$ 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 $\hat{I}“$ 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 $\hat{I}“$.

23 pages, 1 figure