Zimmer’s conjecture for actions of $\mathrm {SL}(m,\pmb {\mathbb {Z}})$

Abstract

We prove Zimmer’s conjecture for $C^2$ actions by finite-index subgroups of $\mathrm {SL}(m,{\mathbb {Z}})$ provided $m>3$. The method utilizes many ingredients from our earlier proof of the conjecture for actions by cocompact lattices in $\mathrm {SL}(m,{\mathbb {R}})$ (Brown et al. in Zimmer’s conjecture: subexponential growth, measure rigidity, and strong property (T), 2016. arXiv:1608.04995) but new ideas are needed to overcome the lack of compactness of the space $(G \times M)/\Gamma$ (admitting the induced G-action). Non-compactness allows both measures and Lyapunov exponents to escape to infinity under averaging and a number of algebraic, geometric, and dynamical tools are used control this escape. New ideas are provided by the work of Lubotzky, Mozes, and Raghunathan on the structure of nonuniform lattices and, in particular, of $\mathrm {SL}(m,{\mathbb {Z}})$ providing a geometric decomposition of the cusp into rank one directions, whose geometry is more easily controlled. The proof also makes use of a precise quantitative form of non-divergence of unipotent orbits by Kleinbock and Margulis, and an extension by de la Salle of strong property (T) to representations of nonuniform lattices.