Diophantine properties of nilpotent Lie groups

Abstract

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 BΓ(n) centered at 1 ∈ Γ never approach the identity of G closer than |BΓ(n)| −β . A Lie group G is said to be Diophantine if for every k ≥ 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.

On metric Diophantine approximation in matrices and Lie groups

“…In a second part of this note, we state new results regarding diophantine approximation on Lie groups, in the spirit of our earlier work [ABRdS14a]. These results, which are applications of the theorems described in the first part of this note, concern the diophantine exponent of nilpotent Lie groups and were our initial motivation for studying diophantine approximation on submanifolds of matrices.…”

“…Inspired by work of Gamburd-Jakobson-Sarnak [GJS99] and BourgainGamburd [BG08] on the spectral gap problem for finitely generated subgroups of compact Lie groups, we defined in a previous article [ABRdS14a] the notion of diophantine subgroup of an arbitrary Lie group G. The definition is as follows. Any finite symmetric subset S := {1,…”

“…Indeed a construction was given in [ABRdS14a] for each integer k ∈ N of a connected Lie group which is diophantine on k letters, but not on k + 1 letters. Our examples are certain nilpotent Lie groups without a rational structure.…”

“…. , e m+n of the s-homogeneous part of the relatively free Lie algebra of G on k generators F k,G (see [ABRdS14a]), the submanifold M k,G of matrices to be considered is the image of (Lie(G)) k under the (polynomial) map sending x ∈ (Lie(G)) k to the (n + m) × m matrix whose columns are the e i (x). Here m = dim G (s) .…”

“…If the k elements s i 's forming S are chosen at random with respect to Haar measure, then α S is almost surely a fixed integer, which is a polynomial in k (see [ABRdS14a]). …”