高秩算术格的渐近生成数

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。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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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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