Separable at Birth: Products of Full Relatively Quasi-Convex Subgroups

Let G be a hyperbolic group such that all quasi-convex subgroups are separable. Minasyan proved that finite products of such subgroups are themselves separable using a combination theorem of Gitik. Mart´ınez-Pedroza and Sisto proved that double cosets of quasi-convex subgroups of a relatively hyperbolic group which have comptible parabolic subgroups are likewise separable. Using the cusped space definition of a relatively hyperbolic group due to Groves and Manning, we prove a combination theorem for full relatively quasi-convex subgroups. Using this theorem, we show that products of full relatively hyperbolic groups are separable if every full relatively quasi-convex subgroup of G is separable.