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

Conjugacy Distinguished Cosets in Hyperbolic -Manifold Groups

A subset of a group is \emph{conjugacy distinguished} if the union of all conjugates of is closed in the profinite topology on . We prove that if is a hyperbolic -manifold of finite volume, , and is an abelian subgroup of , then the coset is conjugacy distinguished in . A subset is \emph{conjugacy distinguished from a class of subgroups} if, for every in the class that is disjoint from the union of conjugates of , there exists a homomorphism , where is a finite group, such that is disjoint from the union of conjugates of . In previous work, we proved that if is a hyperbolic -manifold of finite volume, then a coset of a maximal parabolic subgroup with cusp is conjugacy distinguished from the class of maximal parabolic subgroups of with cusps distinct from . We extend this result by proving that a coset of a loxodromic subgroup is conjugacy distinguished from the class of maximal parabolic subgroups of .