Diophantine properties of nilpotent Lie groups
Aka, Menny ; Breuillard, Emmanuel ; Rosenzweig, Lior ; de Saxcé, Nicolas
In: Compositio Mathematica, 2015, vol. 151, no. 6, p. 1157-1188
Ajouter à la liste personnelle- Summary
- A finitely generated subgroup ${\rm\Gamma}$ of a real Lie group $G$ is said to be Diophantine if there is ${\it\beta}>0$ such that non-trivial elements in the word ball $B_{{\rm\Gamma}}(n)$ centered at $1\in {\rm\Gamma}$ never approach the identity of $G$ closer than $|B_{{\rm\Gamma}}(n)|^{-{\it\beta}}$ . A Lie group $G$ is said to be Diophantine if for every $k\geqslant 1$ 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