On the Asymptotic Number of Generators of High Rank Arithmetic Lattices

Abert, Gelander, and Nikolov [AGR17] conjectured that the number of generators d(Γ) of a lattice Γ in a high rank simple Lie group H grows sublinearly with v = μ(H/)Γ, 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, d(Γ) = O_H (log v/log log 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 d(Γ) = O(log v).