TY - JOUR
T1 - On the Asymptotic Number of Generators of High Rank Arithmetic Lattices
AU - Lubotzky, Alexander
AU - Slutsky, Raz
N1 - Publisher Copyright:
© 2022 University of Michigan. All rights reserved.
PY - 2022/8
Y1 - 2022/8
N2 - 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) = OH (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).
AB - 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) = OH (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).
UR - http://www.scopus.com/inward/record.url?scp=85137384519&partnerID=8YFLogxK
U2 - 10.1307/mmj/20217204
DO - 10.1307/mmj/20217204
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AN - SCOPUS:85137384519
SN - 0026-2285
VL - 72
SP - 465
EP - 477
JO - Michigan Mathematical Journal
JF - Michigan Mathematical Journal
ER -