Let $G$ be a (topologically) simple Hausdorff topological group. Let $H$ be a dense subgroup of $G$. Now throw away the topology. What restrictions are known on the structure of $H$ as an abstract group? I imagine not much can be said if $G$ has a very coarse topology, but I am particularly interested in the case where $G$ is totally disconnected and locally compact, that is, the intersection of all open compact subgroups of $G$ is trivial.


A related question: two (t.d.l.c.) topological groups $G$ and $K$ have a dense subgroup $H$ in common. Suppose $G$ is (topologically) simple. What does this say about $K$?


I don't have a precise question I want to answer here, this is more of an appeal for references on the subject.



Topological simplicity and dense subgroups
In many cases a simple topological group contains a nonabelian free group which is dense. This is true for Lie groups and for many examples of locally compact totally disconnect groups. In many of these cases every limit group can be found as a dense subgroup. Now, I do not rememebr the details, but you can find more informtaion in the webpage of the Workshop on
"The Algebraic Structure of Profinite Groups" see <https://www.ma.rhul.ac.uk/profinite_groups/>. 


