标题： TO_BE_TRANSLATED: Approximate lattices: structure in linear groups, definition(s) and beyond 显示英文标题

标题： Approximate lattices: structure in linear groups, definition(s) and beyond

摘要： TO_BE_TRANSLATED: Approximate lattices are aperiodic generalisations of lattices of locally compact groups first studied in seminal work of Yves Meyer. They are uniformly discrete approximate groups (i.e. 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) come from lattices in higher-dimensional Euclidean spaces via the cut-and-project construction. A fundamental challenge consists in extending Meyer's theorem beyond Euclidean spaces. Our main result provides a complete structure theorem for approximate lattices in linear algebraic groups over local fields and their finite products, in particular generalising Meyer's theorem. Along the way, we extend a theorem of Lubotzky--Mozes--Raghunathan to approximate lattices, we show an arithmeticity statement in perfect groups with a higher-rank condition and build rank one approximate lattices with surprising behaviour. Our work also unveils the role plaid by a novel notion of cohomology for approximate subgroups. The structure of approximate lattices in linear algebraic groups reduces to a cohomology class which we can then prove vanishes in higher-rank building upon a method of Burger--Monod. Beyond approximate lattices, this cohomology is key to prove uniqueness of the quasi-models of approximate groups introduced by Hrushovski. We take this opportunity to collect in one place and one common language the recent advances in the theory of approximate lattices and infinite approximate subgroups of locally compact groups. This work begins with an introduction recording definitions and surveying previous results. In this first part we also tackle the main issue concerning the definition(s) of approximate lattices: there are six competing definitions and little is known of how they relate to one another. 显示英文摘要

摘要： Approximate lattices are aperiodic generalisations of lattices of locally compact groups first studied in seminal work of Yves Meyer. They are uniformly discrete approximate groups (i.e. 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) come from lattices in higher-dimensional Euclidean spaces via the cut-and-project construction. A fundamental challenge consists in extending Meyer's theorem beyond Euclidean spaces. Our main result provides a complete structure theorem for approximate lattices in linear algebraic groups over local fields and their finite products, in particular generalising Meyer's theorem. Along the way, we extend a theorem of Lubotzky--Mozes--Raghunathan to approximate lattices, we show an arithmeticity statement in perfect groups with a higher-rank condition and build rank one approximate lattices with surprising behaviour. Our work also unveils the role plaid by a novel notion of cohomology for approximate subgroups. The structure of approximate lattices in linear algebraic groups reduces to a cohomology class which we can then prove vanishes in higher-rank building upon a method of Burger--Monod. Beyond approximate lattices, this cohomology is key to prove uniqueness of the quasi-models of approximate groups introduced by Hrushovski. We take this opportunity to collect in one place and one common language the recent advances in the theory of approximate lattices and infinite approximate subgroups of locally compact groups. This work begins with an introduction recording definitions and surveying previous results. In this first part we also tackle the main issue concerning the definition(s) of approximate lattices: there are six competing definitions and little is known of how they relate to one another.