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Diophantine properties of nilpotent Lie groups

Part of:Lie groupsLie algebras and Lie superalgebrasProbabilistic theory: distribution modulo $1$; metric theory of algorithms

Published online by Cambridge University Press:  13 January 2015

Menny Aka,
Emmanuel Breuillard,
Lior Rosenzweig and
Nicolas de Saxcé
Menny Aka
Affiliation:
Section de mathématiques, EPFL, Station 8 - Bât. MA, CH-1015 Lausanne, Switzerland email menashe-hai.akkaginosar@epfl.ch
Emmanuel Breuillard
Affiliation:
Laboratoire de Mathématiques, Bâtiment 425, Université Paris Sud 11, 91405 Orsay, France email emmanuel.breuillard@math.u-psud.fr
Lior Rosenzweig
Affiliation:
Department of Mathematics, KTH, SE-100 44 Stockholm, Sweden email lior.rosenzweig@gmail.com
Nicolas de Saxcé
Affiliation:
Einstein Institute of Mathematics, Givat Ram, The Hebrew University, Jerusalem, 91904, Israel email saxce@ma.huji.ac.il
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Abstract

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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.

Keywords

word mapsnilpotent Lie groupsDiophantine subgroupequidistribution

MSC classification

Primary:22E25: Nilpotent and solvable Lie groups11K60: Diophantine approximation
Secondary:17B01: Identities, free Lie (super)algebras

Information

Type
Research Article
Information
Compositio Mathematica , Volume 151 , Issue 6 , June 2015, pp. 1157 - 1188
Copyright
© The Authors 2015 

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