Hausdorff-Gromov Limits of Rescaled Finite Transitive Graphs

Given a finite transitive graph, we consider the underlying metric space obtained by rescaling the graph metric so that the diameter is 1. In this talk we will discuss under which conditions a sequence of such spaces converge for the Hausdorff-Gromov topology and what limit can be achieved. We will take advantage of this interesting question to discuss various aspects of geometric group theory: Cayley graphs, growth function, convergent sequences of metric spaces…The talk is based on a recent work of the speaker joint with Itai Benjamini and Hilary Finucane.