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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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We introduce a notion of cocycle-induction for strong uniform approximate lattices in locally compact second countable groups and use it to relate (relative) Kazhdan- and Haagerup-type of approximate lattices to the corresponding properties of the ambient locally compact groups. Our approach applies to large classes of uniform approximate lattices (though not all of them) and is flexible enough to cover the $L^
We announce results on the structure of CAT(0) groups, CAT(0) lattices and of the underlying spaces. Our statements rely notably on a general study of the full isometry groups of proper CAT(0) spaces. Classical statements about Hadamard manifolds are established for singular spaces; new arithmeticity and rigidity statements are obtained.
Starting with a lattice with an action of $mathbb{Z}$ or $mathbb{R}$, we build a Helly graph or an injective metric space. We deduce that the $ell^infty$ orthoscheme complex of any bounded graded lattice is injective. We also prove a Cartan-Hadamard result for locally injective metric spaces. We apply this to show that any Garside group acts on an injective metric space and on a Helly graph. We also deduce that the natural piecewise $ell^infty$ metric on any Euclidean building of type $tilde{A_n}$ extended, $tilde{B_n}$, $tilde{C_n}$ or $tilde{D_n}$ is injective, and its thickening is a Helly graph. Concerning Artin groups of Euclidean types $tilde{A_n}$ and $tilde{C_n}$, we show that the natural piecewise $ell^infty$ metric on the Deligne complex is injective, the thickening is a Helly graph, and it admits a convex bicombing. This gives a metric proof of the $K(pi,1)$ conjecture, as well as several other consequences usually known when the Deligne complex has a CAT(0) metric.
We show that the group of almost automorphisms of a d-regular tree does not admit lattices. As far as we know this is the first such example among (compactly generated) simple locally compact groups.
We develop the foundations of a geometric theory of countably-infinite approximate groups, extending work of Bjorklund and the second-named author. Our theory is based on the notion of a quasi-isometric quasi-action (qiqac) of an approximate group on a metric space. More specifically, we introduce a geometric notion of finite generation for approximate group and prove that every geometrically finitely-generated approximate group admits a geometric qiqac on a proper geodesic metric space. We then show that all such spaces are quasi-isometric, hence can be used to associate a canonical QI type with every geometrically finitely-generated approximate group. This in turn allows us to define geometric invariants of approximate groups using QI invariants of metric spaces. Among the invariants we consider are asymptotic dimension, finiteness properties, numbers of ends and growth type. A particular focus is on qiqacs on hyperbolic spaces. Our strongest results are obtained for approximate groups which admit a geometric qiqac on a proper geodesic hyperbolic space. For such ``hyperbolic approximate groups we establish a number of fundamental properties in analogy with the case of hyperbolic groups. For example, we show that their asymptotic dimension is one larger than the topological dimension of their Gromov boundary and that - under some mild assumption of being ``non-elementary - they have exponential growth and act minimally on their Gromov boundary. We also study convex cocompact qiqacs on hyperbolic spaces. Using the theory of Morse boundaries, we extend some of our results concerning qiqacs on hyperbolic spaces to qiqacs on proper geodesic metric spaces with non-trivial Morse boundary.
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