Groupes profinis presqu'amalgam\'es

TL;DR

This paper extends Bass-Serre theory to profinite groups by establishing an analogous characterization of amalgamated free products using the theory of prographs, bridging abstract and profinite group actions on trees.

Contribution

It introduces a profinite analogue of Bass-Serre's fundamental theorem using the theory of prographs, providing new insights into profinite group actions.

Findings

Profinite groups can be characterized via actions on protrees.

An analogue of Bass-Serre's theorem is established for profinite groups.

The theory of prographs is effectively applied to profinite group theory.

Abstract

Un des r\'esultats fondamentaux de la th\'eorie de Bass-Serre est le th\'eor\`eme suivant : un groupe est amalgam\'e si et seulement si il agit sur un arbre avec comme domaine fondamental un segment. Dans cet article nous donnons un analogue pour les groupes profinis de ce r\'esultat en utilisant la th\'eorie des prographes de Deschamps et Suarez introduite. A fundamental result of Bass-Serre theory is the following theorem: an abstract group is a free product with amalgamation if and only if it acts on a tree with a segment as fundamental domain. In this article, an analogous result for profinite groups will be given, using the theory of prographs of Deschamps and Suarez