  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.
