Geometry of actions, expanders and warped cones

In this thesis we introduce a notion of graphs approximating actions of finitely generated groups on metric and measure spaces. We systematically investigate expansion properties of said graphs and we prove that a sequence of graphs approximating a fixed action ρ forms a family of expanders if and only if ρ is expanding in measure. This enables us to rely on a number of known results to construct numerous new families of expander (and superexpander) graphs. Proceeding in our investigation, we show that the graphs approximating an action are uniformly quasi-isometric to the level sets of the associated warped cone. The existence of such a relation between approximating graphs and warped cones has twofold advantages: on the one hand it implies that warped cones arising from actions that are expanding in measure coarsely contain families of expanders, on the other hand it provides a geometric model for the approximating graphs allowing us to study the geometry of the expander thus obtained. The rest of the work is devoted to the study of the coarse geometry of warped cones (and approximating graphs). We do so in order to prove rigidity results which allow us to prove that our construction is flexible enough to produce a number of non coarsely equivalent new families of expanders. As a by-product, we also show that some of these expanders enjoy some rather peculiar geometric properties, e.g. we can construct expanders that are coarsely simply connected.