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The tempered fundamental group of a p-adic analytic space classifiescoverings that are dominated by a topological covering (for the Berkovichtopology) of a finite etale covering of the space. Here we constructcospecialization homomorphisms between (p') versions of the temperedfundamental groups of the fibers of a smooth family of curves with semistablereduction. To do so, we will translate our problem in terms of cospecializationmorphisms of fundamental groups of the log fibers of the log reduction. Inparticular, we will have to study invariance of the geometric log fundamentalgroup of saturated log smooth log schemes over a log point by change of logpoint.

Coverings in p-adic analytic geometry and log coverings I: Cospecialization of the (p')-tempered fundamental group for a family of curves