Quasi-Actions on Trees II: Finite Depth Bass-Serre Trees
Quasi-Actions on Trees II: Finite Depth Bass-Serre Trees
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树上的拟作用 II：有限深度 Bass-Serre 树
DOI：
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发表时间：
2004
期刊：
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通讯作者：
K. Whyte
中科院分区：
文献类型：
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作者：
L. Mosher;M. Sageev;K. Whyte
在群图的Bass-Serre树具有有限深度的假设下，研究了有限群图的基群的拟等距刚性和分类问题。群的有限深度图的主要例子是顶点群和边群都是粗糙庞加莱对偶群的群。主要定理表明，在一定的假设下，如果$mathcal{G}$是粗糙庞加莱对偶群的有限图，则$mathcal{G}$的基本群的任何有限生成群的拟等距也是粗糙庞加莱对偶群的有限图的基本群，并且两个这样的群之间的任何拟等距必须粗糙地保持它们的空间的bassserre树的顶点空间和边空间。除了一些简单的归一化假设外，主要的假设是“交叉图条件”，它被施加在每个顶点群$mathcal{G}_v$上，它是一个$n维的粗糙庞加莱对偶群，每个关联边群都有正的余维数：$mathcal{G}_v$的相交图$epsilon_v$描述了$mathcal{G}_v$上的余维数为1的边组与$mathcal{G}_v$上的其他边组相交的模式，相交图条件要求$epsilon_v$连通或为空。
This paper addresses questions of quasi-isometric rigidity and classification for fundamental groups of finite graphs of groups, under the assumption that the Bass-Serre tree of the graph of groups has finite depth. The main example of a finite depth graph of groups is one whose vertex and edge groups are coarse Poincare duality groups. The main theorem says that, under certain hypotheses, if $mathcal{G}$ is a finite graph of coarse Poincare duality groups, then any finitely generated group quasi-isometric to the fundamental group of $mathcal{G}$ is also the fundamental group of a finite graph of coarse Poincare duality groups, and any quasi-isometry between two such groups must coarsely preserve the vertex and edge spaces of their Bass-Serre trees of spaces. Besides some simple normalization hypotheses, the main hypothesis is the "crossing graph condition", which is imposed on each vertex group $mathcal{G}_v$ which is an $n$-dimensional coarse Poincare duality group for which every incident edge group has positive codimension: the crossing graph of $mathcal{G}_v$ is a graph $epsilon_v$ that describes the pattern in which the codimension 1 edge groups incident to $mathcal{G}_v$ are crossed by other edge groups incident to $mathcal{G}_v$, and the crossing graph condition requires that $epsilon_v$ be connected or empty.