The Zimmer Program

The group SL_n(Z) (when n > 2) is very rigid, for example, Margulis proved all its linear representations come from representations of SL_n(R) and are as simple as one can imagine. The Zimmer program states that certain "non-linear" representations (group actions by diffeomorphisms on a closed manifold) come also from basic algebraic constructions. For example, conjecturally the only (non-trivial) action on SL_n(Z) on an (n-1) dimensional manifold is the one on the (n-1) sphere coming projectivizing natural action of SL_n(R) on R^n . I'll describe some recent progress on these questions due to A. Brown, D. Fisher, F. Rodriguez-Hertz, Z. Wang  and myself.