Let be a (topologically) simple Hausdorff topological group. Let be a dense subgroup of . Now throw away the topology. What restrictions are known on the structure of as an abstract group? I imagine not much can be said if has a very coarse topology, but I am particularly interested in the case where is totally disconnected and locally compact, that is, the intersection of all open compact subgroups of is trivial.
A related question: two (t.d.l.c.) topological groups and have a dense subgroup in common. Suppose is (topologically) simple. What does this say about ?
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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