Résume | Björklund and Hartnick recently introduced a type of approximate subgroups called approximate lattices: discrete approximate subgroups of locally compact groups with finite co-volume. Their motivation was to define a non-commutative generalisation of Meyer’s mathematical quasi-crystals (certain aperiodic subsets of Euclidean spaces with long range order). A key question asks whether Meyer’s main theorem, that asserts that quasi-crystals are projections of certain subsets of higher-dimensional lattices, holds true for all approximate lattices.
I will discuss how to relate Meyer’s theorem to a consequence of Hrushovski’s stabilizer theorem and how this idea can be utilised to obtain both an extension of Meyer’s theorem to amenable groups, and a decomposition theorem à la Auslander for approximate lattices in Lie groups. The latter result will then naturally lead us to take a look at approximate lattices in semi-simple groups. While an amenable Meyer-type theorem can be proved by drawing parallels between Meyer’s and Hrushovski’s point of view, we will see how ergodic-theoretic tools from Margulis’ proof of arithmeticity and Zimmer’s proof of cocycle superrigidity lead to a partial solution in the semi-simple case. |