Edge subdivisions and the $L^2$-homology of right-angled Coxeter groups

Abstract

If $L$ is a flag triangulation of $S^{n-1}$, then the Davis complex ${\Sigma }_L$ for the associated right-angled Coxeter group $W_L$ is a contractible $n$-manifold. A special case of a conjecture of Singer predicts that the $L^2$-homology of such ${\Sigma }_L$ vanishes outside the middle dimension. We give conditions which guarantee this vanishing is preserved under edge subdivision of $L$. In particular, we verify Singer's conjecture when $L$ is the barycentric subdivision of the boundary of an $n$-simplex, and for general barycentric subdivisions of triangulations of $S^{2n-1}$. Using this, we construct explicit counterexamples to a torsion growth analogue of Singer's conjecture.

Fixed mistake in the statement and the proof of (old) Theorem 6.1, this is replaced by Theorem 6.1 and Theorem 6.3. All previous parts of the paper are unchanged