Structure of quasiconvex virtual joins

TL;DR

This paper investigates the structure of joins of relatively quasiconvex subgroups in relatively hyperbolic groups, revealing how their intersections with maximal parabolic subgroups behave and establishing a combination theorem for compatible subgroups.

Contribution

It demonstrates that intersections of joins with maximal parabolic subgroups are themselves joins of intersections, and introduces a combination theorem for quasiconvex subgroups with compatible parabolic subgroups.

Findings

Intersections of joins with maximal parabolic subgroups are joins of intersections.

Quasiconvex subgroups with almost compatible parabolics have finite index subgroups with compatible parabolics.

Provides a new combination theorem for such subgroups.

Abstract

Let $G$ be a relatively hyperbolic group and let $Q$ and $R$ be relatively quasiconvex subgroups. It is known that there are many pairs of finite index subgroups $Q' \leqslant_f Q$ and $R' \leqslant_f R$ such that the subgroup join $\langle Q', R' \rangle$ is also relatively quasiconvex, given suitable assumptions on the profinite topology of $G$. We show that the intersections of such joins with maximal parabolic subgroups of $G$ are themselves joins of intersections of the factor subgroups $Q'$ and $R'$ with maximal parabolic subgroups of $G$. As a consequence, we show that quasiconvex subgroups whose parabolic subgroups are almost compatible have finite index subgroups whose parabolic subgroups are compatible, and provide a combination theorem for such subgroups.