Abstract

Approximate lattices are approximate subgroups of locally compact groups that generalise lattices (discrete subgroups of finite co-volume). A theorem due to Yves Meyer asserts that approximate lattices in Euclidean spaces are projections of certain subsets of lattices in higher-dimensional spaces. This raises the following question: does Meyer’s theorem hold for approximate lattices in non-commutative locally compact groups?

I will prove that Meyer’s theorem holds for approximate lattices in amenable locally compact subgroups. To do so I will define “good models” that relate approximate subgroups to compact neighbourhoods of the identity in locally compact groups. I will also explain how similar methods yield a generalisation of Auslander’s theorem about the intersection of a lattice in a Lie group and the amenable radical.