Reconstruction anabélienne du squelette des courbes analytiques

Abstract

In this work we bring to light some anabelian behaviours of analytic curves in the setting of Berkovich geometry. We show more precisely that the knowledge of the tempered fundamental group of some curves that we call analytically anabelian determines their analytic skeletons as graphs. The tempered fundamental group of a Berkovich space, introduced by Andr\'e, enabled Mochizuki to prove the first result of anabelian geometry in Berkovich geometry concerning analytifications of algebraic hyperbolic curves over $\overline{\mathbb{Q}}_p$. To that end, Mochizuki developed the categorical language of semi-graphs of anabelio\ids and tempero\ids. Our work consists in associating a graph of anabelio\ids to a Berkovich curve equipped with a minimal triangulation and in adapting the results of Mochizuki in order to recover the analytic skeleton of the curve. The novelty of this anabelian result in Berkovich geometry is that the curves we are interested in are not supposed anymore to be of algebraic nature. We show for example that the famous Drinfeld half-plane is an analytically anabelian curve.