A quantitative Selberg’s lemma

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 the generalized Riemann hypothesis (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} .