Tempered Fundamental Group and Metric Graph of a Mumford Curve

Abstract

The aim of this paper is to give some general results on the tempered fundamental group of p-adic smooth algebraic varieties (which is a sort of analog of the topological fundamental group of complex algebraic varieties in the p-adic world). The main result asserts that one can recover the metric structure of the graph of the stable model of a Mumford curve from the tempered fundamental group of the curve. We also prove the birational invariance, invariance under algebraically closed extensions and a Kunneth formula for the tempered fundamental group. We describe the tempered fundamental group of an abelian variety and link the tempered fundamental group of a curve to the tempered fundamental group of its Jacobian variety.