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.

ABSTRACT. We introduce and study model sets in commutative spaces, i.e. homogeneous spaces of the form G/K where G is a (typically non-abelian) locally compact group and K is a compact subgroup such that (G, K) is a Gelfand pair. Examples include model sets in hyperbolic spaces, Riemannian symmetric spaces, regular trees and generalized Heisenberg groups. Continuing our work from [6] we associate with every regular model set in G/K a Radon measure on K\G/K called its spherical auto-correlation. We then define the spherical diffraction of the regular model set as the spherical Fourier transform of its spherical auto-correlation in the sense of Gelfand pairs. The main result of this article ensures that the spherical diffraction of a uniform regular model set in a commutative space is pure point. In fact, we provide an explicit formula for the spherical diffraction of such a model set in terms of the automorphic spectrum of the underlying lattice and the underlying window. To describe the coefficients appearing in this formula, we introduce a new type of integral transform for functions on the internal space of the model set. This integral transform can be seen as a shadow of the spherical Fourier transform of physical space in internal space and is hence referred to as the shadow transform of the model set. To illustrate our results we work out explicitly several examples, including the case of model sets in the Heisenberg group.

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We determine the Krieger type of nonsingular Bernoulli actions G g∈G ({0, 1}, µ g ). When G is abelian, we do this for arbitrary marginal measures µ g . We prove in particular that the action is never of type II ∞ if G is abelian and not locally finite, answering Krengel's question for G = Z. When G is locally finite, we prove that type II ∞ does arise. For arbitrary countable groups, we assume that the marginal measures stay away from 0 and 1. When G has only one end, we prove that the Krieger type is always I, II 1 or III 1 . When G has more than one end, we show that other types always arise. Finally, we solve the conjecture of [VW17] by proving that a group G admits a Bernoulli action of type III 1 if and only if G has nontrivial first L 2 -cohomology.

We develop a method for providing quantitative estimates for higher order correlations of group actions. In particular, we establish effective mixing of all orders for actions of semisimple Lie groups as well as semisimple S-algebraic groups and semisimple adele groups. As an application, we deduce existence of approximate configurations in lattices of semisimple groups.