Given a group G and a collection of codimension–one subgroups H of G, one can construct a CAT(0) cube complex on which G acts isometrically with no global fixed point. This is known as Sageev’s construction. In this construction, codimension–one subgroups of H are commensurable with hyperplane stabilizers. By imposing certain conditions on G and H, one can promote the group action to a proper or cocompact one. A proper action is harder to obtain than a cocompact one.

In this talk, we introduce a relative version of Groves–Manning’s result on quasi-convexity in the Sageev construction. Let (G,P) be a finitely generated relatively hyperbolic group acting cocompactly and P-elliptically on a CAT(0) cube complex. In this setting, we show that vertex stabilizers are full relatively quasi-convex if and only if hyperplane stabilizers are full relatively quasi-convex.