On the boundary criterion for relative cubulation: Multiended parabolics

Abstract In this note, we extend the boundary criterion for relative cubulation of the first author and Groves to the case when the peripheral subgroups are not necessarily one‐ended. Specifically, if the boundary criterion is satisfied for a relatively hyperbolic group, we show that, up to taking a refined peripheral structure , the group admits a relatively geometric action on a CAT(0) cube complex. We anticipate that this refinement will be useful for constructing new relative cubulations in a variety of settings.