Local rigidity of affine actions of higher rank groups and lattices


Let $J$ be a semisimple Lie group with all simple factors of real rank at least two. Let $\Gamma\lt J$ be a lattice. We prove a very general local rigidity result about actions of $J$ or $\Gamma$. This shows that almost all so-called “standard actions” are locally rigid. As a special case, we see that any action of $\Gamma$ by toral automorphisms is locally rigid. More generally, given a manifold $M$ on which $\Gamma$ acts isometrically and a torus $\mathbb T^n$ on which it acts by automorphisms, we show that the diagonal action on $\mathbb T^n{\times}M$ is locally rigid.

This paper is the culmination of a series of papers and depends heavily on our work in two recent articles. The reader willing to accept the main results of those papers as “black boxes” should be able to read the present paper without referring to them.


David Fisher

Indiana University
Department of Mathematics
Rawles Hall
Bloomington, IN 47401
United States

Gregory Margulis

Yale University
Department of Mathematics
P.O. Box 208283
New Haven, CT 06520
United States