Abstract Amenability is a notion of facial exposedness for convex cones that stronger than being facially dual complete (or ‘nice’) which is, in turn, merely exposed. Hyperbolicity are family algebraically structured closed contain all spectrahedral (linear sections positive semidefinite cones) as special cases. It known amenable. We establish hyperbolicity As part the argument, we show any fac...

In this paper, we study the geometry of cone-offs CAT(0) cube complexes over a family combinatorially convex subcomplexes, with an emphasis on their Gromov-hyperbolicity. A first application gives direct cubical proof characterization (strong) relative hyperbolicity right-angled Coxeter groups, which is particular case result due to Behrstock, Caprace, Hagen and Sisto. second acylindrical C ′ (...

Open questions 100 1 Appendix. Equivalent definitions of relative hyperbolicity 103 Bibliography 108 2 Chapter 1 Introduction 1.1 Preliminary remarks Originally, the notion of a relatively hyperbolic group was proposed by Gromov [45] in order to generalize various examples of algebraic and geometric nature such as fundamental groups of finite–volume non–compact Riemannian mani-folds of pinched ...