Date of Degree

5-2018

Document Type

Dissertation

Degree Name

Ph.D.

Program

Mathematics

Advisor

Jason Behrstock

Committee Members

Abhijit Champanerkar

Olga Kharlampovich

Subject Categories

Geometry and Topology

Keywords

right-angled Coxeter group, geometric group theory

Abstract

We provide geometric conditions on a pair of hyperplanes of a CAT(0) cube complex that imply divergence bounds for the cube complex. As an application, we characterize right-angled Coxeter groups with quadratic divergence and show right-angled Coxeter groups cannot exhibit a divergence function between quadratic and cubic. This generalizes a theorem of Dani-Thomas that addressed the class of 2-dimensional right-angled Coxeter groups. This characterization also has a direct application to the theory of random right-angled Coxeter groups. As another application of the divergence bounds obtained for cube complexes, we provide an inductive graph theoretic criterion on a right-angled Coxeter group's defining graph which allows us to recognize arbitrary integer degree polynomial divergence for many infinite classes of right-angled Coxeter groups. We also provide similar divergence results for some classes of Coxeter groups that are not right-angled. Finally, we discuss thick structures on right-angled Coxeter groups and show that for n larger than 1, there are right-angled Coxeter groups that are thick of order n but are algebraically thick of strictly larger order, answering a question of Behrstock-Drutu-Mosher.

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