For n 2, consider an action of a lattice in SL(n,R) on a manifold M by C^2 diffeomorphisms. I’ll discuss the following two theorems: (1) if dim(M) n-1 then the action preserves a Borel probability measure; (2) moreover, if the lattice is cocompact then the action factors through the action of a finite group.
A key tool in the proof of either theorem is the use of Lyapunov exponents and metric entropy to obtain invariance of certain probability measures on a related suspension space. I will define the suspension space construction and Lyapunov exponents and formulate this key proposition. I’ll explain how, after some reduction, the above theorems follow. This is joint work with D. Fisher, S. Hurtado, F. Rodriguez Hertz, and Z. Wang.