## Mathematics Ph.D. Dissertations

#### Title

On the quasi-isometric rigidity of a class of right-angled Coxeter groups

#### Date of Award

2019

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy (Ph.D.)

#### Department

Mathematics/Mathematics (Pure)

#### First Advisor

Xiangdong Xie (Advisor)

#### Second Advisor

Maria Bidart (Other)

#### Third Advisor

Kit Chan (Committee Member)

#### Fourth Advisor

Mihai Staic (Committee Member)

#### Abstract

To each finite simplicial graph Γ there is an associated right-angled Coxeter group given by the presentation

WΓ=⟨ v ∈ V(Γ)| v2=1 for all v ∈ V(Γ); v1v2=v2v1 if and only if (v1, v2) ∈ E(Γ)⟩,

where V(Γ),E(Γ) denote the vertex set and edge set of Γ respectively. In this dissertation, we discuss the quasi-isometric rigidity of the class of right-angled Coxeter groups whose defining graphs are given by generalized polygons. We begin with a review of some helpful preliminary concepts, including a discussion on the current state of the art of the quasi-isometric classification of right-angled Coxeter groups. We then prove in detail that for any given joins of finite generalized thick m-gons Γ1,Γ2 with m ∈ {3,4,6,8}, the corresponding right-angled Coxeter groups are quasi-isometric if and only if Γ1 and Γ2 are isomorphic.

#### Recommended Citation

Bounds, Jordan, "On the quasi-isometric rigidity of a class of right-angled Coxeter groups" (2019). *Mathematics Ph.D. Dissertations*. 42.

https://scholarworks.bgsu.edu/math_diss/42