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 $\mathrm{SL}(m,\mathbb{R})$ 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.

中文翻译：

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Zimmer 对 $$\mathrm {SL}(m,\pmb {\mathbb {Z}})$$SL(m,Z) 动作的猜想

我们通过 $\mathrm{SL}(m,\mathbb{Z})$ 提供的 $m>3$ 的有限索引子群证明了 Zimmer 对 $C^2$ 动作的猜想。该方法利用了我们之前对 $\mathrm{SL}(m,\mathbb{R})$ 中协紧格的作用的猜想证明中的许多成分，但是需要新的想法来克服空间 $( G \times M)/\Gamma$（承认诱导的$G$-action）。非紧凑性允许测度和李雅普诺夫指数在平均下逃逸到无穷大，并且使用了许多代数、几何和动力学工具来控制这种逃逸。Lubotzky、Mozes 和 Raghunathan 关于非均匀晶格结构的工作提供了新的想法，特别是 $\mathrm{SL}(m,\mathbb{Z})$ 提供了尖端的几何分解排名一个方向，其几何形状更容易控制。该证明还利用了 Kleinbock 和 Margulis 提出的单能轨道非发散的精确定量形式，以及 de la Salle 对强性质 (T) 的扩展到非均匀晶格的表示。