Structure of quasiconvex virtual joins

Abstract Let be a relatively hyperbolic group and let and be relatively quasiconvex subgroups. It is known that there are many pairs of finite index subgroups and such that the subgroup join is also relatively quasiconvex, given suitable assumptions on the profinite topology of . We show that the intersections of such joins with maximal parabolic subgroups of are themselves joins of intersections of the factor subgroups and with maximal parabolic subgroups of . 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.