The large-scale geometry of right-angled Coxeter groups

Abstract

This is a survey of some aspects of the large-scale geometry of right-angled Coxeter groups. The emphasis is on recent results on their negative curvature properties, boundaries, and their quasi-isometry and commensurability classification. PALLAVI DANIproper cocompact action by the group. We also discuss hierarchical and acylindrical hyperbolicity, which are of great current interest. In Section 4, we discuss visual boundaries of CAT(0) spaces and hyperbolic spaces, and illustrate how rightangled Coxeter groups have played a role in the quest to explore the extent to which properties of boundaries of hyperbolic spaces hold in the CAT(0) setting. We also discuss which topological spaces arise as boundaries of Davis complexes. In recent years there has been an explosion of activity related to the quasi-isometry and commensurability classification of right-angled Coxeter groups. These results are discussed in Sections 5 and 6 respectively.There is a vast body of literature on right-angled Coxeter groups, and the topics presented here are those that are closest to the expertise of the author. Notably missing are the numerous results on algorithmic questions, on various notions of dimension, and on automorphisms and outer automorphisms of right-angled Coxeter groups, as well as the countless instances where right-angled Coxeter groups are used as tools or to produce examples or counterexamples for questions that arise in another area. There is also a budding theory of random right-angled Coxeter groups, and we mention a few results about random groups when they touch upon the topics being discussed.The author would like to thank Ruth Charney and Genevieve Walsh for helpful conversations, Jason Behrstock for comments and help with Section 3.4, and

Quasi-Isometries for certain Right-Angled Coxeter Groups

“…Extending this QI-classification of certain hyperbolic RACGs from [DT17], Hruska, Stark and Tran give a QI-classification for (not necessarily hyperbolic) RACGs whose defining graphs are generalized theta graphs in [HST20, Theorem 1.6]. These results are combined in [Dan18,Theorem 5.20] to a QI-classification of RACGs whose defining graphs are included in the much larger class of graphs dealt with in this paper.…”

Quasi-isometric rigidity of a class of right-angled Coxeter groups

“…Its original construction and definition can be found in [8]. We provide a summary of the construction here as seen in [6].…”

“…Here we give a very brief summary of the current state of the art. For more details see the survey article by Pallavi Dani [6]. By a result of Behrstock-Caprace-Hagen-Sisto [2], the RACGs are divided into two classes, those that are algebraically thick, and those that are relatively hyperbolic.…”