Trees, Ultrametrics, and Noncommutative Geometry

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 Γ on a compact, ultrametric space by local similarities is introduced. It is proved that there is a faithful unitary representation of Γ into 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 O2 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.