三维Coxeter轨道折叠的几何化与Singer猜想
Geometrization of 3-dimensional Coxeter orbifolds and Singer's conjecture
摘要：
与任何Coxeter系统$(W,S)$相关联，都有一个标记单纯复形$L$和一个可收缩CW复形$Σ_L$（戴维斯复形），$W$在该复形上正确地起作用并且是互补的。$Σ_L$允许细胞形成，每个顶点的神经是$L$。因此，如果$L$是$\mathbb{S}^{n-1}$的三角剖分，则$Σ_L$是可收缩的$n$ -流形。在这种情况下，轨道空间$K_L:=Σ_L/W$是一个\emph{Coxeter orbifold}。我们证明了一个类似于三维流形的JSJ-分解的结果：每个三维Coxeter orbifold沿着Euclidean suborbifold分裂成\emph{characteristic suborbifold}和简单（双曲）片。因此，每个三维考克斯特轨道都有一个分解成具有双曲、欧几里得或$\mathbb{H}^2\times\mathbb{R}$几何的片段。(We忽略球形Coxeter orbifolds的情况。）辛格猜想在3维中的一个版本如下：$Σ_L$的约化的$\ell^2$ -同调消失。
英文摘要：
Associated 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.
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