标题： 近似格子在线性群中的结构 显示英文标题

标题： The structure of approximate lattices in linear groups

摘要： 近似格是局部紧群中格的非周期性推广，最初由Yves Meyer在开创性工作中进行了研究。 它们被定义为具有有限余体积的局部紧群中的均匀离散近似子群（对乘法稳定，最多有有限误差的对称子集）。 Meyer表明，欧几里得空间的近似格（也称为Meyer集）通过截断和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定理扩展到欧几里得空间之外。 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，特别是迄今为止对Meyer定理最广泛的扩展。 我们的证明依赖于Lubotzky--Mozes--Raghunathan定理在S-整半单群中的近似格的扩展，一种专门用于研究近似子群的上同调概念，一个补充Hrushovski最近结果的普遍性陈述，以及关于格的有界上同调的Burger和Monod结果的推广。 显示英文摘要

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.