Aspects de la g\'eom\'etrie des groupes

Abstract

This habilitation memoir (in French, submitted in May 2014) is made up of five chapters, each being an introduction to work of the author between 2006 and 2014. The core of the memoir consists of the first three chapters, pertaining to geometric group theory. More precisely, it is concerned with large-scale geometry of groups and especially Lie groups (Chapters 1 and 2), and locally compact hyperbolic groups (Chapter 3). Chapter 4 is concerned with the study of groups through their unitary representations and their isometric actions on Hilbert spaces (Kazhdan and Haagerup properties). Finally, Chapter 5 is more concerned with "structural" group theory, studying, for a given group (discrete, or more generally locally compact), the space of its closed subgroups, or closed normal subgroups. A list of open questions is appended. While the memoir mostly surveys published papers, it also presents some unpublished corollaries in Chapter 1 about asymptotic cones of Lie groups.