Modular index invariants of Mumford curves
type: Item
instance of: scholarly article
zbMATH Open document ID: 1270.46065
author: Matilde Marcolli
author: A. Rennie
author: A. L. Carey
publication date: 2012-08-24T00:00:00Z
full work available at URL: https://arxiv.org/abs/0905.3157
Mathematics Subject Classification ID: 46L85
zbMATH DE Number: 6071193
zbMATH Keywords: Mumford curve
zbMATH Keywords: graph \(C^*\)-algebra
zbMATH Keywords: zhyvot circle action
MaRDI profile type: Publication
arXiv ID: 0905.3157
arXiv classification: math.QA
Recommended article: Q3142387
Recommended article: Q3973868
Recommended article: KMS weights on higher rank buildings
Recommended article: Q3697630
Recommended article: Q3027370
Recommended article: Minimality of Geodesics in Grassmann Manifolds
Recommended article: Hecke algebras from groups acting on trees and HNN extensions.
Recommended article: Green's functions on Mumford curves
Recommended article: \(C^*\)-algebras associated with complex dynamical systems and backward orbit structure
Recommended article: Riemann existence theorems of Mumford type
title: Modular index invariants of Mumford curves
review text: This very interesting paper deals with an interplay between the geometry of algebraic curves over the field of \(p\)-adic numbers (the Mumford curves) and invariants of certain \(C^*\)-algebras attached to such curves.NEWLINENEWLINERecall that, if \(K\) is a finite extension of the field \({\mathbb Q}_p\) of \(p\)-adic numbers, then an algebraic curve \(X(K)\) over \(K\) admits the Mumford uniformization \(X(K)\cong \partial\Delta_K / \Gamma\), where \(\Gamma\subset PGL(2, K)\) is a \(p\)-adic Schottky group and \(\partial\Delta_K\) the space of ends of the Bruhat-Tits tree \(\Delta_K\) associated to \(\Gamma\), see [\textit{D.~Mumford}, Compos. Math. 24, 129--174 (1972; Zbl 0228.14011)]. (The reader can think of \(\Delta_K\) as a \(p\)-adic analog of the hyperbolic upper space \({\mathbb H}^3\) and \(\Gamma\) as an analog of the Fuchsian group acting on \({\mathbb H}^2\cong\partial {\mathbb H}^3\).)NEWLINENEWLINEOne can associate to the infinite (directed) tree \(\Delta_K/\Gamma\) a certain \(C^*\)-algebra \(C^*(\Delta_K/\Gamma)\) known as the graph algebra, see, e.g., [\textit{I.~Raeburn}, Graph algebras. Providence, RI: American Mathematical Society (2005; Zbl 1079.46002)]. Moreover, there exists an element of the group \(\mathrm{Aut}(C^*(\Delta_K/\Gamma))\) called by the authors a zhyvot circle action (`zhyvot' means in Russian `belly' of the graph \(\Delta_K\)).NEWLINENEWLINEThe main result of the paper says that a certain index of the zhyvot action on \(C^*(\Delta_K/\Gamma)\) determines (and is determined by) the length of closed geodesics of the Mumford curve \(X(K)\). A draft of the paper is available at \url{arXiv:0905.3157}.NEWLINENEWLINEFor the entire collection see [Zbl 1243.14002].
reviewed by: Igor V. Nikolaev
description: scientific article; zbMATH DE number 6071193