A Quantitative Selberg's Lemma

We show that an arithmetic lattice $\mathrm{\Gamma }$ in a semi-simple Lie group $G$ contains a torsion-free subgroup of index $\delta (v)$ where $v=\mu (G\mathrm{/}\mathrm{\Gamma })$ 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 $\sim \frac{\log v}{\log \log v}$.