On the Asymptotic Number of Generators of High Rank Arithmetic Lattices

Abstract

Abert, Gelander, and Nikolov [AGR17] conjectured that the number of generators $\mathit{d}(\mathrm{\Gamma })$ of a lattice Γ in a high rank simple Lie group H grows sublinearly with $\mathit{v}=\mathit{\mu }(\mathit{H}/\mathrm{\Gamma })$, the co-volume of Γ in H. We prove this for nonuniform lattices in a very strong form, showing that for 2-generic such Hs, $\mathit{d}(\mathrm{\Gamma })={\mathit{O}}_{\mathit{H}}(log\mathit{v}/loglog\mathit{v})$, which is essentially optimal. Although we cannot prove a new upper bound for uniform lattices, we will show that for such lattices one cannot expect to achieve a better bound than $\mathit{d}(\mathrm{\Gamma })=\mathit{O}(log\mathit{v})$.