Geometrization of 3-dimensional Coxeter orbifolds and Singer's conjecture

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dc.creatorSchroeder, Timothy A.
dc.date2007-10-24
dc.date2008-10-28
dc.date.accessioned2026-07-07T10:13:08Z
dc.date.available2026-07-07T10:13:08Z
dc.descriptionAssociated to any Coxeter system $(W,S)$, there is a labeled simplicial complex $L$ and a contractible CW-complex $Σ_L$ (the Davis complex) on which $W$ acts properly and cocompactly. $Σ_L$ admits a cellulation under which the nerve of each vertex is $L$. It follows that if $L$ is a triangulation of $\mathbb{S}^{n-1}$, then $Σ_L$ is a contractible $n$-manifold. In this case, the orbit space, $K_L:=Σ_L/W$, is a \emph{Coxeter orbifold}. We prove a result analogous to the JSJ-decomposition for 3-dimensional manifolds: Every 3-dimensional Coxeter orbifold splits along Euclidean suborbifolds into the \emph{characteristic suborbifold} and simple (hyperbolic) pieces. It follows that every 3-dimensional Coxeter orbifold has a decomposition into pieces which have hyperbolic, Euclidean, or the geometry of $\mathbb{H}^2\times\mathbb{R}$. (We leave out the case of spherical Coxeter orbifolds.) A version of Singer's conjecture in dimension 3 follows: That the reduced $\ell^2$-homology of $Σ_L$ vanishes.
dc.description15 pages, 3 figures
dc.identifierhttps://arxiv.org/abs/0710.4358
dc.identifierhttp://arxiv.org/abs/0710.4358
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/172422
dc.subjectGroup Theory
dc.subjectAlgebraic Topology
dc.subjectGeometric Topology
dc.subject20F55 (Primary) 20J05, 55N35, 58H10 (Secondary)
dc.titleGeometrization of 3-dimensional Coxeter orbifolds and Singer's conjecture
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