Sur les Automorphismes de Groupes Libres et de Groupes de Surface

TL;DR

This paper explores the analogies between automorphism groups of free groups and surface groups, focusing on their actions on contractible spaces, subgroup properties, and asymptotic geometry.

Contribution

It provides a comparative analysis of the actions and properties of these groups, highlighting similarities and conjectural parallels in their geometric and algebraic structures.

Findings

Identification of common properties of subgroups

Analysis of group actions on contractible spaces

Discussion of conjectural similarities in asymptotic geometry

Abstract

The special linear groups, the mapping class groups of surfaces, the outer autormorphism groups of free groups appear in numerous domains. Their analogies, developped in particular in K. Vogtmann's work, have been written about a lot. In this report, we concentrate on the contractible spaces on which these groups act in an analogous way, on the common properties of their subgroups, and on the similar (or conjecturally similar) properties of their asymptotic geometry.