Global rigidity for totally nonsymplectic Anosov $\mathbb{Z}^k$ actions

Abstract

We consider a totally nonsymplectic (TNS) Anosov action of ${\mathbb{Z}}^k$ which is either uniformly quasiconformal or pinched on each coarse Lyapunov distribution. We show that such an action on a torus is $C^{\infty }$–conjugate to an action by affine automorphisms. We also obtain similar global rigidity results for actions on an arbitrary compact manifold assuming that the coarse Lyapunov foliations are topologically jointly integrable.