On the Asymptotic Number of Generators of High Rank Arithmetic Lattices

TL;DR

This paper proves that for high rank non-uniform lattices in simple Lie groups, the number of generators grows very slowly with the co-volume, confirming a conjecture and providing near-optimal bounds.

Contribution

It establishes the asymptotic growth rate of generators for non-uniform lattices, confirming a conjecture and deriving near-optimal bounds for high rank simple Lie groups.

Findings

Number of generators for non-uniform lattices is O(log v / log log v)

Bounds for uniform lattices are at least O(log v)

Confirms conjecture by Abert, Gelander, and Nikolov

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 $d(\Gamma) = O(\log v)$.