The isomorphism problem for residually torsion-free nilpotent groups. (Q2372586)

In 1911 M. Dehn proposed to decide the following three famous algorithmic problems. Let \(G\) be any group, and suppose that \(W\) is a set of generators for \(G\). The `word problem' asks if there exists a procedure to decide if a word \(w\) on \(W\) represents the identity of \(G\). It was shown by Novikov in 1955 that this problem is in general undecidable. On the other hand, the `conjugacy problem' consists in determining a procedure to decide if two elements \(g_1,g_2\in G\) corresponding to the words \(w_1\) and \(w_2\) on \(W\) are conjugate in \(G\). Finally, the `isomorphism problem' asks if there is a procedure to establish if two groups (given by presentations) are isomorphic. Clearly, as the word problem is undecidable, it follows also the unsolvability of the conjugacy problem and so even that of the isomorphism problem. In 1980 F. Grunewald and D. Segal proved that the conjugacy and isomorphism problems are solvable for finitely generated nilpotent groups, while recently, G. Baumslag showed that the conjugacy problem is (recursively) unsolvable for finitely presented residually torsion-free nilpotent groups. In the article under review the authors complete these researches obtaining, among other results, the following result: There exists a recursive class of presentations of finitely presented residually torsion-free nilpotent groups with unsolvable isomorphism problem.