Résume | One of the fundamental connections between graph theory and group theory goes via the concept of a metric approximation of an
infinite group by finite objects (groups or graphs), one important example being the notion of soficity for groups. This naturally leads
to numerous results which describe approximation properties of the group (for instance, amenability or Haagerup property) in terms of geometric properties of its approximations (e.g. hyperfiniteness or coarse embeddability in a Hilbert space of a graph sequence). As it turns out, ultimately these results rely on analysis of topological and measure-theoretic properties of the coarse groupoid associated to a sequence of graphs of bounded degree. In this talk, I will describe these connections and some recent results around them. |