Oseledets's multiplicative ergodic theorem describes the growth of vectors in R^N under linear ergodic cocycles. I will report on joint work with Anders Karlsson in which we establish an analogous result for mapping class groups of surfaces, describing the various growth rates for the lengths of curves on a hyperbolic surface S under ergodic cocycles of mapping classes of S

This will be an expository talk on the theory of special cube complexes and their application in resolving the virtual Haken conjecture