线性群中近似格的结构
The structure of approximate lattices in linear groups
AI总结：
本文提出了线性群中近似格的完整结构定理，将Meyer定理推广到所有局部域上的线性代数群及其有限积，证明了近似格与格的深层联系。
摘要：
近似格是局部紧群的格的非周期推广，最早由Yves Meyer的开创性工作引入。近似格被定义为局部紧群中具有有限协体积的一致离散近似子群（在有限误差内对乘法封闭的对称子集）。Meyer证明了欧几里得空间的近似格（即Meyer集）通过切割-投影构造与高维欧几里得空间中的格相关联。近似格理论的一个基本挑战是将Meyer定理推广到欧几里得空间之外。\n 我们的主要结果提供了适用于所有局部域上的线性代数群及其有限积的近似格的完整结构定理，特别地，这给出了Meyer定理迄今为止最广泛的推广。我们的证明依赖于将Lubotzky--Mozes--Raghunathan定理推广到S-进半单群中的近似格、一种专为研究近似子群而设计的上同调概念、一个补充Hrushovski近期结果的普适性陈述，以及将Burger和Monod关于格的有界上同调的结果进行推广。
英文摘要：
Approximate lattices are aperiodic generalisations of lattices of locally compact groups that were first studied in seminal work of Yves Meyer. They are defined as those uniformly discrete approximate subgroups (symmetric subsets stable under multiplication up to a finite error) of locally compact groups that have finite co-volume. Meyer showed that approximate lattices of Euclidean spaces (a.k.a. Meyer sets) are related to lattices in higher-dimensional Euclidean spaces via the cut-and-project construction. A fundamental challenge of the theory of approximate lattices is to extend Meyer's theorem beyond Euclidean spaces. Our main result provides a complete structure theorem for approximate lattices valid in all linear algebraic groups over local fields and their finite products, in particular providing the most general extension of Meyer's theorem to date. Our proof relies on an extension of a theorem of Lubotzky--Mozes--Raghunathan to approximate lattices in S-adic semi-simple groups, a notion of cohomology tailored to the study of approximate subgroups, a universality statement complementing a recent result of Hrushovski and a generalisation of a result of Burger and Monod about bounded cohomology of lattices.
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