Modular index invariants of Mumford curves (Q2904906)

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].