Dense Random Finitely Generated Subgroups of Lie Groups

TL;DR

This paper proves that in a connected real Lie group, randomly selecting n+1 elements from a specific neighborhood almost surely generates a dense subgroup, highlighting the typical behavior of subgroup density.

Contribution

It establishes that n+1 randomly chosen elements in a suitable neighborhood almost surely generate a dense subgroup in a connected real Lie group, extending understanding of subgroup generation.

Findings

Randomly chosen n+1 elements generate dense subgroups with probability one.

Existence of a neighborhood where this density property holds.

The result applies to all connected real Lie groups of dimension n.

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.