Introduction to geometric group theory : quasi-isometries and invariants

par Antoine Velut

lundi 17 nov. 2025, 16:30 → 17:30 Europe/Paris

S435 (UMPA)

Viewing groups as geometric objects by making them act on metric spaces : that is the goal of geometric group theory.
In this talk, I aim to give an introduction to this point of view. 

I will start by discussing how many class of groups can be endowed with natural metrics, leading to the notion of Cayley graph of finitely generated groups.

I will then explain how this leads to the study of the large-scale geometry of metric spaces, through the lens of quasi-isometries. I aim to give the proof of the so-called Milnor-Schwarz lemma, which requires very little backgrouund while being very satisfying.

Finally, I will present some quantities that are invariant under quasi-isometries, such as the growth of groups and asymptotic cones. These very intuitive notions capture the idea of studying metric spaces at the large scale, and lead to very strong rigidity results such as Gromov's theorem on groups of polynomial growth.

My goal is to make this accessible to everyone, so there will be as many drawings and intuitive explanations as possible.