On the Asymptotic Number of Generators of High Rank Arithmetic Lattices

2021/01/18 by Lubotzky, Alexander, Slutsky, Raz
#20G30 #22E40 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2101.07227

Abstract

Abert, Gelander and Nikolov [AGN17] conjectured that the number of generators d(Γ) of a lattice Γ in a high rank simple Lie group H grows sub-linearly with v = μ(H / Γ), the co-volume of Γ in H. We prove this for non-uniform lattices in a very strong form, showing that for 2-generic such H's, d(Γ) = O_H(log v / log log v), which is essentially optimal. While we can not prove a new upper bound for uniform lattices, we will show that for such lattices one can not expect to achieve a better bound than d(Γ) = O(log v).

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