Zimmer's conjecture for non-uniform lattices: escape of mass and growth of cocycles

TL;DR

This paper proves Zimmer's conjecture for many higher-rank semisimple Lie groups lattices, addressing challenges related to measure limits and Lyapunov exponents, significantly advancing understanding of low-dimensional group actions.

Contribution

The authors extend Zimmer's conjecture proof to non-uniform lattices, introducing new techniques to control measure escape and Lyapunov exponents in non-compact settings.

Findings

Finiteness of low-dimensional actions established for many higher-rank lattices

New methods developed to prevent escape of mass and Lyapunov exponents

Results significantly improve previous bounds and understanding

Abstract

We establish finiteness of low-dimensional actions of lattices in higher-rank semisimple Lie groups and establish Zimmer's conjecture for many such groups. This builds on previous work of the authors handling the case of actions by cocompact lattices and of actions by $\Sl(n,\Z)$. While the results are not sharp in all cases, they do dramatically improve all known results. The key difficulty overcome in this paper concerns escape of mass when taking limits of sequences of measures. Due to a need to control Lyapunov exponents for unbounded cocycles when taking such limits, quantitative controls on the concentration of mass at infinity are need and novel techniques are introduced to avoid ``escape of Lyapunov exponent."