Nic Brody: Geometry of Coxeter groups.

Abstract

There are many exciting areas of research in the combinatorial properties of Coxeter groups, but that's not what this talk is about. I will review some fundamental concepts of geometric group theory, and sketch the proofs that Coxeter groups are CAT(0) and cubulated. I will describe an "alternative" that the class of Coxeter groups satisfy: a Coxeter group either contains a surface subgroup or has a finite index free subgroup. After this, I'll briefly describe several recent advances in understanding the geometry of Coxeter groups, including the theory of limit roots, the twist conjecture, virtual specialness, and more.
Although many of these concepts may be unfamiliar to you, I'll try to keep the talk example-laden, and describe how many of these properties correspond to basic combinatorial properties of the defining graph of W.