Abert, Gelander, and Nikolov [AGR17] conjectured that the number of generators of a lattice Γ in a high rank simple Lie group H grows sublinearly with , 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, , 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 .
Dedicated to Gopal Prasad on his 75th birthday
"On the Asymptotic Number of Generators of High Rank Arithmetic Lattices." Michigan Math. J. 72 465 - 477, August 2022. https://doi.org/10.1307/mmj/20217204