math.GT daily digest: 1 new submission for 25 June 2026

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 is a hyperbolic $3$-manifold of finite volume, , and $H$ is an abelian subgroup of $\Gamma$, then the coset $gH$ is conjugacy distinguished in $\Gamma$. A subset 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 , where $F$ is a finite group, such that $\phi (K)$ is disjoint from the union of conjugates of $\phi (S)$. In previous work, we proved that if 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 $\Gamma$.