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, Hagen and Sisto. A second application gives the acylindrical hyperbolicity of C ′ (1/4)-T(4) small cancellation quotients of free products.
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