English

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

Dynamical Systems 2020-07-14 v4 Differential Geometry Group Theory Geometric Topology

Abstract

We prove several cases of Zimmer's conjecture for actions of higher-rank cocompact lattices on low dimensional manifolds. For example, if Γ\Gamma is a cocompact lattice in Sl(n,R)\mathrm{Sl}(n, \mathbb R), MM is a compact manifold, and ω\omega a volume form on MM we show that any homomorphism ρ ⁣:ΓDiff(M)\rho\colon \Gamma \rightarrow \mathrm{Diff}(M) has finite image if the dimension of MM is less than n1n-1 and that any homomorphism ρ ⁣:ΓDiff(M,ω)\rho\colon \Gamma \rightarrow \mathrm{Diff}(M,\omega) has finite image if the dimension of MM is less than nn. The key step in the proof is to show 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 (T) to establish the existence of an invariant Riemannian metric.

Keywords

Cite

@article{arxiv.1608.04995,
  title  = {Zimmer's conjecture: Subexponential growth, measure rigidity, and strong property (T)},
  author = {Aaron Brown and David Fisher and Sebastian Hurtado},
  journal= {arXiv preprint arXiv:1608.04995},
  year   = {2020}
}

Comments

Revised and improved for referee comments