A Quantitative Selberg's Lemma

2023/11/27 by Gelander, Tsachik, Slutsky, Raz
#20G30 #20H05 #22E40 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2311.15976

Abstract

We show that an arithmetic lattice Γ in a semi-simple Lie group G contains a torsion-free subgroup of index δ(v) where v = μ(G/Γ) is the co-volume of the lattice. We prove that δ 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 ∼ (log v)/(log log v).

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