  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."
