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arXiv · 1212.4215

Statements and Dilemmas Regarding the $\ell^2$-homology of Coxeter groups

Timothy A Schroeder
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Abstract

We generalize the methods in previous work to provide a program for proving Singer's Conjecture for Coxeter systems. Specifically, we consider even Coxeter systems with nerves that are flag triangulations of $\BS^{n-1}$, $n=2k$. We prove that Singer's Conjecture in dimensions $n-2$ and $n-1$, along with the vanishing of the $\ltwo$-homology of certain subspaces called "two-letter" ruins above dimension $k+1$, imply Singer's Conjecture in dimension $n$. This is, so far, an incomplete program. The author intends this paper to serve as a reference for those inquiring about Singer's Conjecture and about even Coxeter systems.

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Timothy A Schroeder. 2012-12-18. Statements and Dilemmas Regarding the $\ell^2$-homology of Coxeter groups. https://arxiv.org/abs/1212.4215

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