树、超度量与非交换几何
Trees, ultrametrics, and noncommutative geometry
摘要：
非交换几何用于研究超度量空间的局部几何和无穷远处树的几何。康纳斯关于彭罗斯镶嵌的非对易空间的例子被解释为紧致超度量空间在其局部等距群作用下的非豪斯多夫轨道空间。这被推广到紧的，局部刚性的，超度量空间。这些空间中的局部等距型和局部相似型可以用广群C^*-代数来分析。引入了可数群Γ在紧超距离空间上的局部相似局部刚性作用的概念。证明了在作用的芽广群C^*-代数中存在一个忠实的Γ的酉表示.典型的例子是超度量康托尔集上的汤普森群V的标准作用。在这种情况下，C^*-代数是Cuntz代数O_2，并且恢复了最初由Birget和Nekrashevych提出的表示。树的端空间是超度量空间的来源。一些连接之间的局部刚性，超度量空间和一个概念的理论树格的巴斯和Lubotzky。
英文摘要：
Noncommutative geometry is used to study the local geometry of ultrametric spaces and the geometry of trees at infinity. Connes's example of the noncommutative space of Penrose tilings is interpreted as a non-Hausdorff orbit space of a compact, ultrametric space under the action of its local isometry group. This is generalized to compact, locally rigid, ultrametric spaces. The local isometry types and the local similarity types in those spaces can be analyzed using groupoid C^*-algebras. The concept of a locally rigid action of a countable group Γon a compact, ultrametric space by local similarities is introduced. It is proved that there is a faithful unitary representation of Γinto the germ groupoid C^*-algebra of the action. The prototypical example is the standard action of Thompson's group V on the ultrametric Cantor set. In this case, the C^*-algebra is the Cuntz algebra O_2 and representations originally due to Birget and Nekrashevych are recovered. End spaces of trees are sources of ultrametric spaces. Some connections are made between locally rigid, ultrametric spaces and a concept in the theory of tree lattices of Bass and Lubotzky.
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