Diophantine properties of nilpotent Lie groups

TL;DR

This paper characterizes Diophantine properties of finitely generated subgroups in nilpotent Lie groups, establishing criteria based on Lie algebra laws and identifying classes of nilpotent groups that are Diophantine.

Contribution

It provides a characterization of Diophantine nilpotent Lie groups via their Lie algebra laws and identifies specific classes that are Diophantine, advancing understanding in this area.

Findings

Nilpotent Lie groups of class ≤ 5 are Diophantine.

Nilpotent Lie groups of derived length ≤ 2 are Diophantine.

Rational nilpotent Lie groups are Diophantine.

Abstract

A finitely generated subgroup {\Gamma} of a real Lie group G is said to be Diophantine if there is \beta > 0 such that non-trivial elements in the word ball B_\Gamma(n) centered at the identity never approach the identity of G closer than |B_{\Gamma} (n)|^{-\beta}. 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.