On the geometric and algebraic structure of hierarchically hyperbolic groups

Abstract

This thesis is concerned with hierarchically hyperbolic spaces. Firstly we provide an introduction to the theory following an expository approach and highlighting the main ideas and fundamental theorems. We will present a survey of the most useful tools that appear in the literature, in order to provide a source that can be used for future reference. Subsequently, we will then focus on various notions of convexity in hierarchically hyperbolic spaces. We show the equivalence among several, usually distinct, properties, such as median con- vexity, contracting properties and strong quasiconvexity. Using this equivalence, we show that the hyperbolically embedded subgroups of hierarchically hyperbolic groups are precisely those that are almost malnormal and strongly quasiconvex. We also obtain new insight on the geometry of several known examples. While many commonly studied hierarchically hyperbolic spaces have the property that that every strongly quasiconvex subset is either hyperbolic or coarsely covers the entire space, right-angled Coxeter groups exhibit a wide variety of strongly quasiconvex subsets. Finally, we focus on uniform exponential growth in the class of virtually torsion-free hierarchi- cally hyperbolic groups. We provide several sufficient conditions for uniform exponential growth. Notable examples are acylindrical hyperbolicity or the presence of an asymptotic cone with a cut- point. Our methods give a new unified proof of uniform exponential growth for several examples of groups with notions of non-positive curvature. In particular, we obtain the first proof of uni- form exponential growth for certain groups that act geometrically on CAT(0) cubical complexes of dimension 3 or more. Under additional hypotheses, we show that a quantitative Tits alternative holds for hierarchically hyperbolic groups.