Envelopes of certain solvable groups

In this article we introduce and study uniform and non-uniform approximate lattices in locally compact second countable (lcsc) groups. These are approximate subgroups (in the sense of Tao) which simultaneously generalize lattices in lcsc group and mathematical quasi-crystals (a.k.a. Meyer sets) in lcsc abelian groups.We show that envelopes of strong approximate lattices are unimodular, and that approximate lattices in nilpotent groups are uniform. We also establish several results relating properties of approximate lattices and their envelopes. For example, we prove a version of the Milnor-Schwarz lemma for uniform approximate lattices in compactly-generated lcsc groups, which we then use to relate metric amenability of uniform approximate lattices to amenability of the envelope.Finally we extend a theorem of Kleiner and Leeb to show that the isometry groups of higher rank symmetric spaces of non-compact type are QI rigid with respect to finitely-generated approximate groups.

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“…Explicitly it appears e.g. in [23,18]. Given f ∈ QI(X) we denote by [f ] the equivalence class of f in QI(X).…”

Section: The Left-regular Quasi-actionmentioning

confidence: 99%

Approximate lattices

Björklund¹,

Hartnick²

2018

Duke Math. J.

170

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“…the groups that can contain Γ as a lattice, is very natural since lattices generally reflect the properties of the ambient group. This problem is addressed in [Dym15] for certain solvable groups, and structure results of envelopes of a large class of countable groups have been announced in [BFS14]. Note that the groups G(F, F ′ ) that are finitely generated have infinite amenable commensurated subgroups, and therefore do not satisfy the assumptions of [BFS14].…”

Section: Simplicitymentioning

confidence: 99%

Groups acting on trees with almost prescribed local action

Boudec¹

2016

Comment. Math. Helv.

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We investigate a family of groups acting on a regular tree, defined by prescribing the local action almost everywhere. We study lattices in these groups and give examples of compactly generated simple groups of finite asymptotic dimension (actually one) not containing lattices. We also obtain examples of simple groups with simple lattices, and we prove the existence of (infinitely many) finitely generated simple groups of asymptotic dimension one. We also prove various properties of these groups, including the existence of a proper action on a CAT(0) cube complex.

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“…Additionally we show how Theorem 1.2 can be used to simplify some of the proofs of quasi-isometric rigidity found in [6] and [18,19]. Then we show how both Theorem 1.2 and Theorem 5.4 can be used to improve results on envelopes of abelian-by-cyclic groups found in [8]. Finally in Theorems 5.8 and 5.9 we prove results on quasi-isometric rigidity of Lie groups and locally compact groups quasi-isometric to certain solvable Lie groups.…”

Section: Applications To Quasi-isometric Rigiditymentioning

confidence: 90%

“…In [11], Furman classifies all locally compact envelopes of lattices in semisimple Lie groups and outlines a technique using the quasi-isometry group to solve Problem 5.1 for cocompact lattice embeddings. In [8] the first author adapts this outline for cocompact lattices to solve the envelopes problem for various classes of solvable groups. Below we show how Theorems 1.2 and 5.4 simplify the proofs from [8] in certain cases and extend the results to other groups as well.…”

Section: 4mentioning

confidence: 99%