The Poisson boundary of CAT(0) cube complex groups
The Poisson boundary of CAT(0) cube complex groups
复制标题
CAT(0) 立方复群的泊松边界
DOI：
10.4171/ggd/202
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发表时间：
2011
期刊：
影响因子：
--
通讯作者：
M. Sageev
中科院分区：
文献类型：
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作者：
A. Nevo;M. Sageev
我们考虑一个有限维的、局部有限的CAT(0)立方复形\(X\)，它允许一个余紧的、真不连续的可数自同构群\(G\)。我们构造一个自然的紧度量空间\(B(X)\)，\(G\)通过同胚作用在其上，该作用是极小的且强近距的。此外，对于\(G\)上的任何生成概率测度，\(B(X)\)都存在唯一的平稳测度，并且当该测度具有有限对数矩时，它构成泊松边界的一个紧度量模型。我们确定了\(B(X)\)的一个稠密的\(G_\delta\)子集，\(G\)在其上的作用是波莱尔顺从的，并描述了这两个空间与罗勒边界的关系。我们的构造可用于给出该复形的性质\(A\)的一个简单几何证明。我们的方法基于关于半空间的渐近行为及其极限超滤子的直接几何论证，这些论证具有相当大的独立研究价值。特别地，我们分析了复形中的中位数和区间的概念，并通过\(V. 凯曼诺维奇\)发展的条带准则，在证明\(B(X)\)是泊松边界时使用了后者。
We consider a finite-dimensional, locally finite CAT(0) cube complex X admitting a co-compact properly discontinuous countable group of automorphisms G. We construct a natural compact metric space B(X) on which G acts by homeomorphisms, the action being minimal and strongly proximal. Furthermore, for any generating probability measure on G, B(X) admits a unique stationary measure, and when the measure has finite logarithmic moment, it constitutes a compact metric model of the Poisson boundary. We identify a dense G-delta subset of B(X) on which the action of G is Borel-amenable, and describe the relation of these two spaces to the Roller boundary. Our construction can be used to give a simple geometric proof of Property A for the complex. Our methods are based on direct geometric arguments regarding the asymptotic behavior of half-spaces and their limiting ultrafilters, which are of considerable independent interest. In particular we analyze the notions of median and interval in the complex, and use the latter in the proof that B(X) is the Poisson boundary via the strip criterion developed by V. Kaimanovich.