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.
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