Infinite approximate subgroups of soluble Lie groups

Simon, Machado,

doi:10.1007/s00208-021-02258-8

Cited by 2 publications

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References 70 publications

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“…We now prove the technical lemma (Lemma 2.9) used earlier in the proof of Lemma 2.8. It will be deduced easily from a combination of a generalisation of Schreiber's theorem to the solvable setup [Mac22] and results concerning approximate subgroups contained in neighbourhoods of normal amenable subgroups [Mac23b].…”

Section: Wrapping Up the Proofmentioning

confidence: 99%

“…By the Ado-Iwasawa theorem [Hoc65, Proof of XVIII.3.2], there is a faithful representation ρ : L → GL n (R) such that ρ(A) is unipotent, hence closed. So the map ρ V A is proper for any symmetric relatively compact neighbourhood of the identity V in L. According to [Mac22], there is a closed connected subgroup N of GL n (R) normalised by ρ(L) and a compact subset C such that ρ(Λ sol ) ⊂ CN and N ⊂ Cρ(Λ sol ). We want to be able to choose N ⊂ ρ(A).…”

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Approximate Lattices in Higher-Rank Semi-Simple Groups

Simon

2023

Geom. Funct. Anal.

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Approximate lattices of locally compact groups were first studied in a seminal monograph of Yves Meyer and were subsequently used in the theory of aperiodic order to model objects such as Pisot numbers, quasi-cristals or aperiodic tilings. Meyer studied approximate lattices of Euclidean spacesnow dubbed Meyer sets. A fascinating feature of this theory is the wealth of natural and equivalent definitions of Meyer sets that are available.The study of approximate lattices in non-commutative groups has recently attracted more attention. They are meant as generalisation of lattices of locally compact groups (i.e. discrete subgroup of finite co-volume) and, as such, are roughly defined as approximate subgroups of locally compact groups that are both discrete and have finite co-volume. In this generalized framework, the correct notion of 'finite co-volume' is however up for interpretation. A number of definitions recently arose following work of Björklund and Hartnick, Hrushovski, and the author. In stark contrast with the Euclidean framework, little is known of the relations between these notions.Our main objective in this paper is to address this issue. We establish clear relations between these definitions and show that, in fact, these definitions are not all equivalent. We prove nevertheless that all the notions mentioned are related to one another and simply represent varying degrees of generality. Among the original results we prove here, we extend a cornerstone theorem of Meyer and show that all strong approximate lattices are laminar; we build measures with weak invariance properties on the invariant hull of approximate lattices; we use quasi-models to show that any BH-approximate lattice is contained in an approximate lattice; we prove that even laminar approximate lattices can be arbitrarily far from strong approximate lattices and model sets.

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Section: Wrapping Up the Proofmentioning

confidence: 99%

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confidence: 99%