On dense free subgroups of Lie groups -- revisited

TL;DR

This paper proves that dense subgroups of connected Lie groups can be generated by a small number of elements and provides a detailed quantitative analysis of contracting projective transformations.

Contribution

It establishes that every dense subgroup of a connected Lie group contains a dense subgroup generated by twice the dimension of the group, with detailed proofs on contraction quantification.

Findings

Dense subgroups contain a 2d-generated dense subgroup, where d=dim(G)

Provides a quantitative characterization of contracting projective transformations

Enhances understanding of subgroup generation in Lie groups

Abstract

We show that every dense subgroup of a connected Lie group G contains a dense subgroup generated by 2d elements, where d=dim(G). We also give a detailed proof for the quantitive characterization of a contracting projective transformation in terms of the ratio between the two leading terms in its Cartan decomposition.