Given a non-linear action of a discrete group on a torus, one can always construct a linearized action by toral automorphisms. We ask under what conditions do these actions coincide after a change of coordinates. With F. Rodriguez Hertz and Z. Wang, we show in a recent paper that for actions of SL(n,Z), n>= 3, under a suitable lifting hypothesis and assuming the action on homology is hyperbolic, such a change of coordinates exists intertwining the non-linear and linear actions (when restricted to finite index subgroups).

I will explain the construction of the linearized action, the statement of our main theorems, and indicate the main ideas in our proof. Speaker(s): Aaron Brown (University of Chicago)