Smooth Infinitesimal Rigidity for Higher Rank Partially Hyperbolic Actions on Step-2 Nilmanifolds
If Λ is finitely generated and M is compact, an action φ M → M is a C ∞ homomorphism : Λ ×from Λ to Diff(M ). There is a natural formal tangent space at the point [φ] determined by φ, which is given by the 1-cocycles over φ with coefficients in the smooth vector fields on M . The 1-coboundaries form a closed subspace of the formal tangent space, and when these two spaces are equal, the action is said to be infinitesimally rigid. The purpose of this thesis is to use representation theory to prove the infinitesimal rigidity of partially hyperbolic actions on a family of 2-step free nilmanifolds. We start by characterizing the irreducible representations in L2 (Γ\N ) using the coadjoint orbit method. Then we introduce the obstructions to solving the twisted coboundary equation λω − ω ◦ A = θ, and prove how these obstructions vanish for the whole action due to the higher rank condition.