Dense subgroups of Lie Groups 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!
 A: You might also be interested in a theorem of Breuillard, Gelander, Souto and Storm.  They prove that a connected, semisimple Lie group contains a dense copy of any finitely generated, fully residually free group (a 'limit group' in Sela's terminology).  The reference is
Breuillard, Gelander, Souto, Storm, Dense embeddings of surface groups.
Geom. Topol. 10 (2006), 1373--1389.
The class of limit groups is large but very well understood, and includes fundamental groups of most surfaces.
A: I recently completed a preprint with Michael Larsen in this direction. Here is a link to the paper:
https://arxiv.org/abs/1312.7294
Here is the abstract:
When does Borel's theorem on free subgroups of semisimple groups generalize to other groups? We initiate a systematic study of this question and find positive and negative answers for it. In particular, we fully classify fundamental groups of surfaces and von Dyck groups that satisfy Borel's theorem. Further, as a byproduct of this theory, we make headway on a question of Breuillard, Green, Guralnick, and Tao concerning double word maps.
