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August 2022 On the Asymptotic Number of Generators of High Rank Arithmetic Lattices
Alexander Lubotzky, Raz Slutsky
Michigan Math. J. 72: 465-477 (August 2022). DOI: 10.1307/mmj/20217204

Abstract

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(logv/loglogv), 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(logv).

Dedication

Dedicated to Gopal Prasad on his 75th birthday

Citation

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Alexander Lubotzky. Raz Slutsky. "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

Information

Received: 14 January 2021; Revised: 6 May 2021; Published: August 2022
First available in Project Euclid: 2 August 2022

Digital Object Identifier: 10.1307/mmj/20217204

Subjects:
Primary: 22E40
Secondary: 20G30

Rights: Copyright © 2022 The University of Michigan

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Vol.72 • August 2022
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