Coning-off CAT(0) cube complexes

TL;DR

This paper investigates the geometry of cone-offs of CAT(0) cube complexes, providing new insights into their hyperbolic properties and applications to group theory, including hyperbolicity characterizations and small cancellation quotients.

Contribution

It offers a cubical proof of relative hyperbolicity for right-angled Coxeter groups and establishes acylindrical hyperbolicity for certain small cancellation quotients.

Findings

Characterization of hyperbolicity of cone-offs of CAT(0) cube complexes.

Cubical proof of relative hyperbolicity for right-angled Coxeter groups.

Proving acylindrical hyperbolicity of specific small cancellation quotients.

Abstract

In this paper, we study the geometry of cone-offs of CAT(0) cube complexes over a family of combinatorially convex subcomplexes, with an emphasis on their Gromov-hyperbolicity. A first application gives a direct cubical proof of the characterization of the (strong) relative hyperbolicity of right-angled Coxeter groups, which is a particular case of a result due to Behrstock, Caprace and Hagen. A second application gives the acylindrical hyperbolicity of $C'(1/4)-T(4)$ small cancellation quotients of free products.