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

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