Trees, Ultrametrics, and Noncommutative Geometry

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 ty...

Noncommutative Geometry on Trees and Buildings

The notion of a spectral triple, introduced by Connes (cf. [9], [7], [10]), provides a powerful generalization of Riemannian geometry to noncommutative spaces. It originates from the observation that, on a smooth compact spin manifold, the infinitesimal line element ds can be expressed in terms of the inverse of the classical Dirac operator D, so that the Riemannian geometry is entirely encoded...

ar X iv : m at h / 06 05 13 1 v 2 [ m at h . O A ] 1 1 Ju n 20 07 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 ty...

ar X iv : m at h / 06 05 13 1 v 1 [ m at h . O A ] 4 M ay 2 00 6 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 ty...

Noncommutative geometry