AN ABSTRACT OF THE DISSERTATION OF Anton Kaul for the degree Doctor of Philosophy in Mathematics presented on May 3, 2000. Title: Rigidity for a Class of Coxeter Groups Abstract approved:

approved: William A. Bogley A Coxeter group W is said to be rigid if, given any two Coxeter systems (W,S) and (W,S ′), there is an automorphism ρ of W which carries S to S ′ In this dissertation we review rigidity results of D. Radcliffe [29] and of R. Charney and M. Davis [11], noting that certain restrictions must be placed on the Coxeter group in order to obtain rigidity. Radcliffe examined “right-angled” Coxeter groups, while Charney and Davis considered Coxeter groups of “type HMn” (both of which are defined in Section 4.2). We introduce Coxeter groups of “typeKn,” which may be viewed as a natural generalization of type HM2 Coxeter groups. Relying primarily on techniques of geometric group theory, we show that if W is Coxeter group of type Kn with Coxeter systems (W,S) and (W,S ′), then the associated Coxeter graphs ΓS and ΓS′ are isomorphic. With added restrictions on the system (W,S) this is sufficient to conclude that W is rigid. The geometric analysis mentioned above is performed on a simplicial complex Σ, called the the Davis complex, which was introduced by M. Davis in [13]. G. Moussong [24] showed that the Davis complex Σ supports a CAT (0) metric dM (called the Moussong metric) and that the Coxeter group W acts properly and cocompactly by isometries on the metric space (Σ, dM). This geometric structure enables one to deduce a wealth of information about the Coxeter group. We observe that passing from type HM2 Coxeter groups groups to type Kn Coxeter groups introduces greater topological complexity in the Davis complex Σ. We undertake a study of certain subcomplexes of Σ with the goal of alleviating the complications. In doing so we obtain a topological proof that if r ∈ W is a reflection and W is of type Kn, then the centralizer CW (r) is isomorphic to Z2 × F, where F is a free group. For general Coxeter groups, a similar result has been obtained by B. Brink [9] using different methods. c ©Copyright by Anton Kaul May 3, 2000 All Rights Reserved Rigidity for a Class of Coxeter Groups