[Paper Review] Dense Random Finitely Generated Subgroups of Lie Groups

J. Winkelmann|ArXiv.org|Sep 7, 2003

Geometry and complex manifolds5 references3 citations

TL;DR

This paper proves that for any connected real Lie group $G$ of dimension $n$, choosing $n+1$ elements at random from a sufficiently small relatively compact open neighborhood $W$ of the identity yields a subgroup that is dense in $G$ with probability one. The proof relies on properties of Zassenhaus neighborhoods, regular elements, and the structure of quotients by normal subgroups, showing that dense generation occurs generically under mild topological conditions.

ABSTRACT

Let G be a connected real Lie group of dimension n. Then there exists a relatively compact open neighbourhood W of e in G such that for n+1 randomly chosen elements g_0,..,g_n the generated subgroup will be dense in G with probability one.

Motivation & Objective

• To establish conditions under which randomly chosen elements from a Lie group generate a dense subgroup.
• To determine the minimal number of generators required for dense subgroups in connected real Lie groups.
• To analyze the role of Zassenhaus neighborhoods and regular elements in ensuring dense generation.
• To compare results for nilpotent, perfect, and general Lie groups, highlighting structural differences in random generation.

Proposed method

• Uses Zassenhaus neighborhoods—open, relatively compact subsets where iterated commutators converge to identity—to ensure local group behavior suitable for probabilistic analysis.
• Applies the concept of regular elements in Lie groups, whose complement has Haar measure zero, to avoid pathological cases in random generation.
• Employs a reduction argument via quotients: if the image of a subgroup is dense modulo a normal subgroup $N$, and $N$ is contained in the subgroup, then the subgroup is dense in $G$.
• Leverages the fact that for a regular element $g$ in a connected Lie group $G$, any connected closed normal subgroup $H$ containing $g$ forces $G/H$ to be nilpotent.
• Uses measure-theoretic arguments in $\mathbb{R}^n$ to show that $n+1$ randomly chosen vectors generate a dense subgroup if their coefficients are $\mathbb{Q}$-linearly independent.
• Applies results from Breuillard and Gelander on perfect Lie groups to show that fewer generators (e.g., 2 for semisimple groups) suffice in special cases.

Experimental results

Research questions

• RQ1 What is the minimal number of randomly chosen elements needed to generate a dense subgroup in a connected real Lie group?
• RQ2 Under what conditions on a neighborhood $W$ of the identity does $n+1$ random elements in $W$ generate a dense subgroup with probability one?
• RQ3 How do the properties of regular elements and Zassenhaus neighborhoods contribute to the genericity of dense subgroups?
• RQ4 Why is $n+1$ the optimal number of generators for general Lie groups, and can this be improved for special classes like nilpotent or perfect groups?
• RQ5 Can dense subgroups be generated with fewer than $n+1$ elements in certain Lie groups, and if so, under what structural conditions?

Key findings

• For any connected real Lie group $G$ of dimension $n$, there exists a relatively compact open neighborhood $W$ of the identity such that $n+1$ randomly chosen elements from $W$ generate a dense subgroup in $G$ with probability one.
• The result is optimal in the sense that $n+1$ generators are necessary for $({\mathbb{R}}^n, +)$, as fewer than $n+1$ elements cannot generate a dense subgroup.
• The neighborhood $W$ can be taken as a Zassenhaus neighborhood, but not every relatively compact open neighborhood suffices—there exist counterexamples where $n$ random elements fail to generate a dense subgroup with positive probability.
• For nilpotent Lie groups, the number of generators needed for dense generation may be less than $\dim G + 1$, and in some cases, $k$-generated subgroups with $k < \dim G + 1$ are neither dense nor discrete with probability one.
• For perfect Lie groups, especially semisimple ones, fewer generators suffice: two elements suffice for dense generation due to results of Breuillard and Gelander, showing that the $n+1$ bound is not tight in this case.
• The set of tuples generating non-dense subgroups has Haar measure zero in $G^{n+1}$, confirming that dense generation is a generic phenomenon under the given conditions.