SETUP: Let $G$ be a connected Lie group, and $H\subset G$ be a FINITELY GENERATED dense subgroup.

I am interested in knowing what kind of information one can infer on the complexity of $H$.

I am especially interested in the case in which $G$ is simply connected, non compact, and non diffeomorphic to $\mathbb{R}^n$. After some research online, the only result I found in this direction is in "On dense free subgroups of Lie groups", by Breuillard, E. and Gelander, T.. Here the authors prove that if $G$ is not solvable, and $H\subset G$ is finitely generated and dense, then it contains a free group of rank $r=2\dim G$.

Does anyone have other references of result in this direction? I hope to find results of the type "such a group $H$ needs to be at least this complicated".

In the case I am interested in, $H$ is the fundamental group of a compact manifold, so I have an "upper bound" on the complexity of "H". Now I want a "lower bound", if this makes any sense.

Thank you in advance!