Let $\Gamma, \Lambda\subset G$ be countably infinite subgroups of a common compact subgroup G. I am interested in properties that one would have to inherit from the other (ie if $\Gamma$ has this property then so does $\Lambda$.)
One initial thing that I thought was that you have a left-right action of $\Gamma\times\Lambda$ on $G$ with the Haar measure. Thus from this you might be able to induce unitary representation from $\Gamma$ to $\Lambda$ much like you would do with measure equivalence and hopefully from this follows some things that we know to be measure equivalence invariants (amenability, property T, etc.) Though I haven't actually verified that this works. Beyond this I can only think of abelian. I would be particularly interested in the case that G is the profinite completion of both $\Gamma$ and $\Lambda$.
Alternatively, are there examples where $\Gamma$ and $\Lambda$ can be quite different.