Zimmer’s conjecture: Subexponential growth, measure rigidity, and strong property (T)

Abstract

We prove several cases of Zimmer’s conjecture for actions of higher-rank, cocompact lattices on low-dimensional manifolds. For example, if $\Gamma$ is a cocompact lattice in $\mathrm{SL}(n,\mathbb{R})$, $M$ is a compact manifold, and $\omega$ a volume form on $M$, we show that any homomorphism $\alpha : \Gamma \rightarrow \mathrm{Diff}(M)$ has finite image if the dimension of $M$ is less than $n-1$ and that any homomorphism $\alpha : \Gamma \rightarrow \mathrm{Diff}(M,\omega)$ has finite image if the dimension of $M$ is less than $n$. The key step in the proof is to show that any such action has uniform subexponential growth of derivatives. This is established using ideas from the smooth ergodic theory of higher-rank abelian groups, structure theory of semisimple groups, and results from homogeneous dynamics. Having established uniform subexponential growth of derivatives, we apply Lafforgue’s strong property \rm (T) to establish the existence of an invariant Riemannian metric.