Diophantine properties of nilpotent Lie groups

Abstract

A finitely generated subgroup Γ of a real Lie group G is said to be Diophantine if there is β> 0 such that non-trivial elements in the word ball B_Γ(n) centered at the identity never approach the identity of G closer than |B_Γ (n)|^{-β}. A Lie group G is said to be Diophantine if for every k > 0, a random k-tuple in G generates a Diophantine subgroup. Semi-simple Lie groups are conjectured to be Diophantine but very little is proven in this direction. We give a characterization of Diophantine nilpotent Lie groups in terms of the ideal of laws of their Lie algebra. In particular we show that nilpotent Lie groups of class at most 5, or derived length at most 2, as well as rational nilpotent Lie groups are Diophantine. We also find that there are non Diophantine nilpotent and solvable (non nilpotent) Lie groups.

33 pages. To appear in Compositio Mathematica. Added Theorem 1.4., which constructs examples, for each integer k, of a Lie group that is diophantine on k letters but not on k+1 letters