I will introduce a construction that given an action on a compact space outputs a metric space. Dynamical and ergodic properties of the action correspond to quasi-isometry invariants of that metric space (called the warped cone).
I will explain how this can be used to construct a continuum of super-expanders comparable with these of Lafforgue and will sketch the proof of a conjecture of Druţu and Nowak that such spaces yield new counterexamples to the coarse Baum--Connes conjecture. | 
