Higher Rank Lattices Research Articles

Affine actions of higher rank lattices

We consider actions of lattices in certain higher rank simple Lie groups by affine (i.e. connection-preserving) transformations of a compact Riemannian manifold. When the dimension of the manifold is not too large, such actions are partially described here in terms of affine actions on the flat torus and isometric actions. The main tools are Marguils' and Zimmer's rigidity theorems.

Rigidity for Anosov Actions of Higher Rank Lattices

This is the simplest example of a large class of analytic actions of lattices in semisimple Lie groups on locally homogeneous spaces. A basic problem is to understand the differentiable actions near to such a standard action in terms of their geometry and dynamics (cf. [13], [50], [51]). A Cr-action So: IF > X -x X of a group F on a compact manifold X is said to be Anosov if there exists at least one element, Yh E F, such that 'p(Yh) is an Anosov diffeomorphism of X. We begin in this paper to study the Anosov differentiable actions of lattices, including many standard algebraic examples, and especially to study their properties. Our main theme is that either the Cr-rigidity or the Cr-deformation rigidity of an Anosov action (for 1 < r < cr. or even for the real analytic case) can be shown just by studying the behavior of the periodic orbits for the action. There are two notions of structural stability that appear in this paper, rigidity and deformation rigidity. A Cl-perturbation of a Cr-action So is simply another Cr-action Spl such that for a finite set of generators {81, . . , ad of F, the Cr-diffeomorphisms 'P(8i) and 'Pl(8i) are Cl-close for all i. An action So is said to be Cr-rigid (or topologically rigid if r = 0) if every sufficiently small C'perturbation of So is Cr-conjugate to So, for 0 < r < oo, or r = w in the case of real analytic actions. A Cl-deformation of an action So is a continuous path of Cr-actions (Pt defined for some 0 < t < e with Spo = SD* An action SD is said to be Cr-deform-