Rigidity of High Dimensional Graph Manifolds
Rigidity of High Dimensional Graph Manifolds
复制标题
高维图流形的刚性
DOI：
10.24033/ast.938
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发表时间：
2011
期刊：
影响因子：
--
通讯作者：
A. Sisto
中科院分区：
文献类型：
--
作者：
R. Frigerio;J.;A. Sisto
我们定义了一类高维图流形。它们是紧致光滑流形，支持分解成有限多个片断，每个片断都微分同胚于一个环面和一个具有环面尖点的有限体积双曲流形的乘积。不同的部分通过边界圆环的仿射地图连接在一起。我们要求块中的所有双曲因子至少有3维。我们的主要目的是从刚性理论的观点来研究这类图流形。
我们证明了，在高维空间中，Borel猜想对我们的图流形成立。我们还证明了光滑刚性在这类中成立：两个图流形是同伦等价的当且仅当它们是微分同胚的。我们引入了不可约图流形的概念，形成了一个具有较好粗几何性质的子类。我们建立了与不可约图流形的基本群拟等距的有限生成群的一些结构理论：任何这样的群都有一个对边群和顶点群有强约束的分裂群的图。最后，我们证明了在每一维>；3中都存在不支持任何局部CAT(0)度量的不可约图流形的例子。
We define the class of high dimensional graph manifolds. These are compact smooth manifolds supporting a decomposition into finitely many pieces, each of which is diffeomorphic to the product of a torus with a finite volume hyperbolic manifold with toric cusps. The various pieces are attached together via affine maps of the boundary tori. We require all the hyperbolic factors in the pieces to have dimension at least 3. Our main goal is to study this class of graph manifolds from the viewpoint of rigidity theory.
We show that, in high dimensions, the Borel conjecture holds for our graph manifolds. We also show that smooth rigidity holds within the class: two graph manifolds are homotopy equivalent if and only if they are diffeomorphic. We introduce the notion of irreducible graph manifolds, which form a subclass which has better coarse geometric properties. We establish some structure theory for finitely generated groups which are quasi-isometric to the fundamental group of an irreducible graph manifold: any such group has a graph of groups splitting with strong constraints on the edge and vertex groups. Finally, we prove that in every dimension >3 there exist examples of irreducible graph manifolds which do not support any locally CAT(0) metric.
影响因子：
1.8
作者：
Behrstock J
通讯作者：
Behrstock J