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This paper is an attempt to give some general results on the temperedfundamental group of a $p$-adic smooth algebraic varieties (which is a sort ofanalog of the topologic fundamental group of complex algebraic varieties in thep-adic world). The main result asserts that one can recover the metricstructure of the graph of the stable model of a Mumford curve from the temperedfundamental group of this Mumford curve. We will also prove birationalinvariance, invariance by algebraically closed extensions and a Kuennethformula for the tempered fundamental group. We will describe the temperedfundamental group of an abelian variety and link the tempered fundamental groupof a curve to the tempered fundamental group of its Jacobian variety.

Tempered fundamental group and metric graph of a Mumford curve