On the Asymptotic Number of Generators of High Rank Arithmetic Lattices

Abstract

Abert, Gelander and Nikolov [AGN17] conjectured that the number of generators $d(\Gamma )$ of a lattice $\Gamma$ in a high rank simple Lie group $H$ grows sub-linearly with $v=\mu (H/\Gamma )$, the co-volume of $\Gamma$ in $H$. We prove this for non-uniform lattices in a very strong form, showing that for $2-$generic such $H$'s, $d(\Gamma )=O_H(\log v/\log \log v)$, 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 $d(\Gamma )=O(\log v)$.