On the Quasi-Isometric Rigidity of a Class of Right-Angled Coxeter Groups

ON THE QUASI-ISOMETRIC RIGIDITY OF A CLASS OF RIGHT-ANGLED COXETER GROUPS

Jordan Bounds

A Dissertation

Submitted to the Graduate College of Bowling Green State University in partial fulﬁllment of the requirements for the degree of

DOCTOR OF PHILOSOPHY

August 2019

Committee:

Xiangdong Xie, Advisor

Maria Bidart, Graduate Faculty Representative

Kit Chan

Xiangdong Xie, Advisor

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

2 WΓ = hv ∈ V (Γ)|v = 1 for all v ∈ V (Γ); v1v2 = v2v1 if and only if (v1, v2) ∈ E(Γ)i, 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. iv ACKNOWLEDGMENTS

I would like to thank Drs. Bidart, Chan, Staic, and Xie for participating in my dissertation committee. I greatly appreciate the time and effort they devoted to my examination. In particular, Dr. Xie has dedicated much of his time to provide me with years of valuable instruction and guidance. My experiences under his tutelage have played signiﬁcant roles in my preparation for a future in academia and I am forever grateful for his efforts. I would also like to thank the Department of Mathematics and Statistics at Bowling Green as a whole for supporting me in my doctoral pursuits. The opportunities afforded to me by the department included a stellar education and immersed me in a community of strong mathematical professionals and students. Thank you all for the countless helpful conversations about math, life, and football. Finally, I would like to thank my family. My parents who have always supported me and continued to sacriﬁce to give me every opportunity that they could; my wife who stood by me while completing her own education. I am forever in their debt and will continue to do all that I can to make them proud. v

TABLE OF CONTENTS Page

CHAPTER 1 PRELIMINARIES ...... 1 1.1 Finitely generated groups ...... 2 1.2 Cayley graphs ...... 6 1.3 Quasi-isometries ...... 8

CHAPTER 2 SPACES OF NONPOSITIVE CURVATURE ...... 15 2.1 Geodesics ...... 15

n 2.2 The model space Xκ ...... 15 2.3 CAT(κ) spaces ...... 17 2.4 Gromov hyperbolic spaces ...... 18 2.5 Gromov’s link condition ...... 19

CHAPTER 3 RIGHT-ANGLED COXETER GROUPS ...... 22 3.1 Background in right-angled Coxeter groups ...... 22 3.2 Ends ...... 26 3.3 Divergence and thickness ...... 29

CHAPTER 4 QUASI-ISOMETRIC RIGIDITY OF A CLASS OF RIGHT-ANGLED COX- ETER GROUPS ...... 32 4.1 Fuchsian buildings ...... 33 4.2 Generalized polygons ...... 35 4.3 The Davis complex as a labeled 2-complex ...... 37 4.4 Proof of main results ...... 43 vi CHAPTER 5 CONSTRUCTING COMMENSURABLE RACGS ...... 46 5.1 Commensurability ...... 46 5.2 A construction of commensurable RACGs ...... 47

BIBLIOGRAPHY ...... 54 vii

LIST OF FIGURES Figure Page

2 1.1 Cayley graph of Z (left) and Cayley graph of F2 (right)...... 7

3.1 Finite example of Davis complex ...... 24

3.2 The Davis complex associated to D∞ ...... 25

4.1 Thick generalized 3-gon...... 36

4.2 Labeling of 4-cycle in PL...... 39

5.1 Commensurable construction when Γ is a 5-cycle ...... 51 viii