Structure of quasiconvex virtual joins

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.