高秩算术格的渐近生成数

IF 0.8 3区数学 Q2 MATHEMATICS

Michigan Mathematical Journal Pub Date : 2021-01-18 DOI:10.1307/mmj/20217204

A. Lubotzky, Raz Slutsky

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引用次数: 4

摘要

Abert, Gelander和Nikolov [AGN17]推测，高阶单李群H中晶格Γ的生成子d(Γ)的数量随着H中的协同体积(Γ) v = μ(H/Γ)呈次线性增长，我们以非常强的形式证明了这一点，表明对于2−一般的H, d(Γ) = OH(log v/ log log v)，这是本质上最优的。虽然我们不能证明一致格的新上界，但我们将证明，对于这样的格，我们不能期望获得比d(Γ) = O(log v)更好的上界。

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On the Asymptotic Number of Generators of High Rank Arithmetic Lattices

Abert, Gelander and Nikolov [AGN17] conjectured that the number of generators d(Γ) of a lattice Γ in a high rank simple Lie group H grows sub-linearly with v = μ(H/Γ), the co-volume of Γ in H. We prove this for non-uniform lattices in a very strong form, showing that for 2−generic such H’s, d(Γ) = OH(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(Γ) = O(log v).

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来源期刊

Michigan Mathematical Journal 数学-数学

CiteScore

1.20

自引率

11.10%

发文量

审稿时长

>12 weeks

期刊介绍： The Michigan Mathematical Journal is available electronically through the Project Euclid web site. The electronic version is available free to all paid subscribers. The Journal must receive from institutional subscribers a list of Internet Protocol Addresses in order for members of their institutions to have access to the online version of the Journal.