Local rigidity of affine actions of higher rank groups and lattices

TL;DR

This paper establishes a broad local rigidity result for actions of higher rank semisimple Lie groups and their lattices, showing that most standard actions are locally rigid, including actions by toral automorphisms.

Contribution

It proves a general local rigidity theorem for actions of higher rank groups and lattices, extending known results to a wide class of actions and settings.

Findings

Almost all standard actions are locally rigid.

Actions by toral automorphisms are locally rigid.

Diagonal actions on product manifolds are locally rigid.

Abstract

Let $J$ be a semisimple Lie group with all simple factors of real rank at least two. Let $\Gamma<J$ 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 $\Ta^n$ on which it acts by automorphisms, we show that the diagonal action on $\Ta^n{\times}M$ is locally rigid. This paper is the culmination of a series of papers and depends heavily on our work in \cite{FM1,FM2}. 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.