Abstract
We extend classical density theorems of Borel and Dani–Shalom on lattices in semisimple, respectively solvable algebraic groups over local fields to approximate lattices. Our proofs are based on the observation that Zariski closures of approximate subgroups are close to algebraic subgroups. Our main tools are stationary joinings between the hull dynamical systems of discrete approximate subgroups and their Zariski closures.
Highlights
Borel’s density theorem [6] is a cornerstone of the theory of lattices in semisimple algebraic groups over local fields, and can be stated as follows.THEOREM 1 (Borel Density Theorem)
A similar density theorem for lattices in solvable algebraic groups was established by Dani [8] and c The Author(s) 2019
Commensurability defines an equivalence relation on subsets of G, and as a first application of stationary hull joinings we show that the class of approximate lattices is invariant under commensurability
Summary
Borel’s density theorem [6] is a cornerstone of the theory of lattices in semisimple algebraic groups over local fields, and can be stated as follows. Using recurrence properties of unipotents on projective space with respect to the invariant measure at hand one deduces that G/H must be a point This approach does not apply directly to our more general setting for several reasons: Firstly, the Zariski closure of an approximate lattice Λ ⊂ G is not a group. Throughout this article, we use the following convention: if k is a local field and G is a linear algebraic group over k, all topological terms (for example, closure, compactness) concerning subsets of G := G(k) refer to the Hausdorff topology on G and not to the Zariski topology, unless explicitly mentioned otherwise
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