$C^1$ actions on manifolds by lattices in Lie groups.

In this paper we study Zimmer's conjecture for $C^1$ actions of lattice subgroup of a higher-rank simple Lie group with finite center on compact manifolds. We show that when the rank of the lattice is larger than the dimension of the manifold, then the action factors through a finite group. In particular, we prove Zimmer's conjecture for $C^1$ actions of lattices in $SL(n, \R)$.