Propri\'et\'es combinatoires du bord d'un groupe hyperbolique

TL;DR

This survey explores combinatorial modulus and Loewner properties as innovative tools for studying quasi-conformal boundary properties of hyperbolic groups, offering new insights into rigidity questions and recent applications.

Contribution

It introduces combinatorial modulus and Loewner properties as novel methods for analyzing boundary quasi-conformal structures of hyperbolic groups, advancing rigidity theory.

Findings

Presented combinatorial modulus and Loewner property as new analytical tools.

Connected these tools to rigidity results in hyperbolic groups.

Demonstrated applications through recent research outcomes.

Abstract

The goal of this survey is to present combinatorial modulus that have been recently used to study quasi-conformal properties of boundaries of hyperbolic groups. First, we will recall well known rigidity results and questions that motivated the introduction of these tools. Then we will define combinatorial modulus and the combinatorial Loewner property that provide new approaches to rigidity questions. Finally, we will describe some applications of these tools through recent results. ----- Le but de ce survol est de pr\'esenter les modules combinatoires r\'ecemment utilis\'es pour \'etudier les propri\'et\'es quasi-conformes des bords des groupes hyperboliques. Dans un premier temps, on rappellera quelques r\'esultats et questions de rigidit\'e bien connus qui ont motiv\'es l'introduction de ces outils. Puis on d\'efinira les modules combinatoires et la propri\'et\'e de L\"owner combinatoire qui offrent une nouvelle approche pour r\'esoudre des probl\`emes ouverts depuis longtemps. Enfin, on d\'ecrira des applications concr\`etes de ces outils \`a travers quelques r\'esultats r\'ecents.