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We prove the Zorich–Kontsevich conjecture that the non-trivial Lyapunov exponents of the Teichmüller ow on (any connected component of a stratum of) the moduli space of Abelian differentials on compact Riemann surfaces are all distinct. By previous work of Zorich and Kontsevich, this implies the existence of the complete asymptotic Lagrangian flag describing the behavior in homology of the vertical foliation in a typical translation surface.
We study property (T) and the fixed-point property for actions on Lp and other Banach spaces. We show that property (T) holds when L2 is replaced by Lp (and even a subspace/quotient of Lp), and that in fact it is independent of 1≤p<∞. We show that the fixed-point property for Lp follows from property (T) when 1< p< 2+ε. For simple Lie groups and their lattices, we prove that the fixed-point property for Lp holds for any 1< p<∞ if and only if the rank is at least two. Finally, we obtain a superrigidity result for actions of irreducible lattices in products of general groups on superreflexive spaces.
We use microlocal and paradifferential techniques to obtain L8 norm bounds for spectral clusters associated with elliptic second-order operators on two-dimensional manifolds with boundary. The result leads to optimal Lq bounds, in the range 2⩽q⩽∞, for L2 - normalized spectral clusters on bounded domains in the plane and, more generally, for two-dimensional compact manifolds with boundary. We also establish new sharp Lq estimates in higher dimensions for a range of exponents q̅n⩽q⩽∞.