  We show that an arithmetic lattice $\Gamma$ in a semi-simple Lie group $G$
contains a torsion-free subgroup of index $\delta(v)$ where $v = \mu
(G/\Gamma)$ is the co-volume of the lattice. We prove that $\delta$ is
polynomial in general and poly-logarithmic under GRH. We then show that this
poly-logarithmic bound is almost optimal, by constructing certain lattices with
torsion elements of order $\sim \frac{\log v}{\log \log v}$.
