Aut\`omats de Stallings, un cam\'i d'anada i tornada

TL;DR

This paper reviews the fundamental properties of free groups and provides a detailed account of Stallings's automata theory, highlighting its geometric interpretation of subgroups and its significance in understanding classical results and generating new ones.

Contribution

It offers a comprehensive review of Stallings's automata theory and its applications to free groups, emphasizing its geometric perspective and recent developments.

Findings

Enhanced understanding of free group properties

Detailed exposition of Stallings's automata theory

Illustration of the theory's applications to classical and new results

Abstract

In this paper we review some of the fundamental properties of the free group and give a detailed account of Stallings's theory of automata, a geometric interpretation of its subgroups that has been (and still is) immensely fruitful, both as a means of understanding classical results, as as a source of new results. We review some of the most important ones. -- - En aquest article revisem algunes de les propietats fonamentals del group lliure i fem una exposici\'o detallada de la teoria dels aut\`omats de Stallings, una interpretaci\'o geom\`etrica dels seus subgrups que ha estat (i segueix essent) immensament fruct\'ifera, tant com a mitj\`a per entendre resultats cl\`assics, com com a font de nous resultats. N'expliquem alguns dels m\'es rellevants.