The $ell^2$-homology of even Coxeter groups

Given a Coxeter system (W,S), there is an associated CW-complex, Sigma, on which W acts properly and cocompactly. We prove that when the nerve L of (W,S) is a flag triangulation of the 3-sphere, then the reduced $\ell^2$-homology of Sigma vanishes in all but the middle dimension.

💡 Research Summary

The paper studies the ℓ²‑homology of Davis complexes associated to Coxeter groups, focusing on the case where the Coxeter system (W,S) is even and its nerve L is a flag triangulation of the 3‑sphere. In this setting Σ(W,S) is a 4‑dimensional contractible CW‑complex on which W acts properly and cocompactly; consequently Σ is a topological 4‑manifold. The main result (Theorem 1.3) asserts that the reduced ℓ²‑homology of Σ vanishes in every dimension except the middle one, i.e. H_i(Σ)=0 for i≠2.

The proof proceeds by introducing a special subspace of Σ called the (S,t)‑ruin, denoted Ω, together with its boundary ∂Ω. The ruin is built from the subgroup W_U, where U={s∈S | m_{st}<∞} for a fixed generator t∈S. Ω decomposes into finitely many “boundary collars” of the form B×