Quasiconvexity of virtual joins and separability of products in relatively hyperbolic groups

TL;DR

This paper proves that in certain relatively hyperbolic groups, the join of finitely generated relatively quasiconvex subgroups remains relatively quasiconvex and that products of such subgroups are closed in the profinite topology, leading to subgroup separability results.

Contribution

It establishes the quasiconvexity of joins and the separability of products of finitely generated relatively quasiconvex subgroups in relatively hyperbolic groups.

Findings

Join of finitely generated relatively quasiconvex subgroups is relatively quasiconvex.

Products of finitely generated relatively quasiconvex subgroups are closed in the profinite topology.

Results apply to limit groups, Kleinian groups, and certain fundamental groups.

Abstract

A relatively hyperbolic group $G$ is said to be QCERF if all finitely generated relatively quasiconvex subgroups are closed in the profinite topology on $G$. Assume that $G$ is a QCERF relatively hyperbolic group with double coset separable (e.g., virtually polycyclic) peripheral subgroups. Given any two finitely generated relatively quasiconvex subgroups $Q,R \leqslant G$ we prove the existence of finite index subgroups $Q'\leqslant_f Q$ and $R' \leqslant_f R$ such that the join $\langle Q',R'\rangle$ is again relatively quasiconvex in $G$. We then show that, under the minimal necessary hypotheses on the peripheral subgroups, products of finitely generated relatively quasiconvex subgroups are closed in the profinite topology on $G$. From this we obtain the separability of products of finitely generated subgroups for several classes of groups, including limit groups, Kleinian groups and balanced fundamental groups of finite graphs of free groups with cyclic edge groups.