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Diophantine properties of nilpotent Lie groups
- Part of
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- Journal:
- Compositio Mathematica / Volume 151 / Issue 6 / June 2015
- Published online by Cambridge University Press:
- 13 January 2015, pp. 1157-1188
- Print publication:
- June 2015
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- Article
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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.
