For polycyclic groups, a result of Grunewald, Pickel and Segal (Polycyclic groups with isomorphic finite quotients. Ann. of Math. (2) 111 (1980), no. 1, 155--195.) says that the class of f.g. polycyclic groups with the same profinite completion (and even with the same sets of finite homomorphic images) is finite. Thus at least for polycyclic groups one can say whether the profinite completions are different just by looking at the groups. Already for metabelian groups the situation is quite different (see Pickel, P. F. Metabelian groups with the same finite quotients. Bull. Austral. Math. Soc. 11 (1974), 115--120. ). There are also some results about free solvable groups. In general, I do not think this (very interesting) question has been studied enough.
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For polycyclic groups, a result of Grunewald, Pickel and Segal (Polycyclic groups with isomorphic finite quotients. Ann. of Math. (2) 111 (1980), no. 1, 155--195.) says that the class of f.g. polycyclic groups with the same profinite completion (and even with the same sets of finite homomorphic images) is finite. Thus at least for polycyclic groups one can say whether the profinite completions are different just by looking at the groups. There are also some results about free solvable groups. In general, I do not think this (very interesting) question has been studied enough. In particular, I do not know the answer to Simon Thomas' question above. |
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For polycyclic groups, a result of Grunewald, Pickel and Segal (Polycyclic groups with isomorphic finite quotients. Ann. of Math. (2) 111 (1980), no. 1, 155--195.) says that the class of f.g. polycyclic groups with the same profinite completion (and even with the same sets of finite homomorphic images) is finite. Thus at least for polycyclic groups one can say whether the profinite completions are different just by looking at the groups. There are also some results about free solvable groups. In general, I do not think this (very interesting) question has been studied enough. In particular, I do not know the answer to Simon Thomas' question above. |
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