On the boundary criterion for relative cubulation: multi-ended parabolics

TL;DR

This paper extends the boundary criterion for relative cubulation to cases with multi-ended parabolics, enabling the construction of new relatively geometric actions on CAT(0) cube complexes.

Contribution

It shows that the boundary criterion applies even when peripheral subgroups are not one-ended, with a refined peripheral structure ensuring relative cubulation.

Findings

Boundary criterion applies to multi-ended parabolics

Refined peripheral structure enables relative cubulation

Facilitates new constructions of relative cubulations

Abstract

In this note, we clarify that the boundary criterion for relative cubulation of the first author and Groves works even when the peripheral subgroups are not 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.