Theses and Dissertations

May 2015

Dissertation

Degree Name

Doctor of Philosophy

Mathematics

Boris L. Okun

Committee Members

Ric Ancel, Craig Guilbault, Chris Hruska, Allen Bell

Keywords

Coxeter Groups, Fattened Davis Complex, L^2 Cohomology, Singer Conjecture

Abstract

Associated to a Coxeter system $(W,S)$ there is a contractible simplicial complex $\Sigma$ called the Davis complex on which $W$ acts properly and cocompactly by reflections. Given a positive real multiparameter $\Q$ indexed by $S$, one can define the weighted $L^2$--(co)homology groups of $\Sigma$ and associate to them a nonnegative real number called the weighted $L^2$--Betti number. Unfortunately, not much is known about the behavior of these groups when $\Q$ lies outside a certain restricted range, and weighted $L^2$--Betti numbers have proven difficult to compute. We propose a program to compute the weighted $L^2$--(co)homology of $\Sigma$ by introducing a thickened version of this complex which we call the fattened Davis complex. A salient feature of this complex is that our construction produces a homology manifold with boundary possessing $\Sigma$ as a $W$--equivariant retract. This allows us to use many standard algebraic topology tools such as Poincar\'{e} duality for computing the $L^2$--(co)homology of $\Sigma$, and we successfully perform computations for many examples of Coxeter groups.

Within the spectrum of weighted $L^2$--(co)homology there is a conjecture of interest called the Weighted Singer Conjecture. The conjecture claims that if $\Sigma$ is an $n$--manifold (equivalently, the nerve of the corresponding Coxeter group is an $(n-1)$--sphere), then the weighted $L^2$--(co)homology groups of $\Sigma$ vanish above dimension $\frac{n}{2}$ whenever $\Q\leq\mathbf{1}$. We present a proof of the conjecture in dimension three that encompasses all but nine Coxeter groups. Then, under some restrictions on the nerve of the Coxeter group, we obtain partial results whenever $n=4$ (in particular, the conjecture holds for $n=4$ if the nerve of the corresponding Coxeter group is a flag complex). We also prove a version of this conjecture in dimensions three and four whenever $\Sigma$ is a manifold with (nonempty) boundary, and then extend our results in dimension four to prove a general version of the conjecture for the case where the nerve of the Coxeter group is assumed to be a flag triangulation of a $3$--manifold.

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