Relatively quasiconvex subgroups in relatively hyperbolic groups form an important class of subgroups and play a similar role to quasiconvex subgroups in word hyperbolic groups. Given two relatively quasiconvex subgroups $Q$ and $R$, the intersection $Q \cap R$ is again relatively quasiconvex, but the join $\langle Q,R\rangle$ may not be. In my talk I will discuss conditions ensuring the existence of finite index subgroups $Q' \leqslant_f Q$ and $R' \leqslant_f R$ such that $\langle Q',R' \rangle$ is relatively quasiconvex. This tool will then be applied to show that products of relatively quasiconvex subgroups are closed in the profinite topology on the ambient relatively hyperbolic group.
The talk will be based on joint work with Lawk Mineh.
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