a1 Department of Mathematics, Fairfield University, Fairfield, Connecticut 06430, U.S.A.
a2 Fachbereich Mathematik Universitat, Dortmund, 44221 Dortmund, Federal Republic of Germany
a3 Fachbereich Mathematik Universitat, Dortmund, 44221 Dortmund, Federal Republic of Germany
In 1962 Gilbert Baumslag introduced the class of groups Gi, j for natural numbers i, j, defined by the presentations Gi, j = < a, b, t; a−1 = [bi, a] [bj, t] >. This class is of special interest since the groups are para-free, that is they share many properties with the free group F of rank 2.
Magnus and Chandler in their History of Combinatorial Group Theory mention the class Gi, j to demonstrate the difficulty of the isomorphism problem for torsion-free one-relator groups. They remark that as of 1980 there was no proof showing that any of the groups Gi, j are non-isomorphic. S. Liriano in 1993 using representations of Gi, j into PSL(2, pk), k , showed that G1,1 and G30,30 are non-isomorphic. In this paper we extend these results to prove that the isomorphism problem for Gi, 1, i is solvable, that is it can be decided algorithmically in finitely many steps whether or not an arbitrary one-relator group is isomorphic to Gi, 1. Further we show that Gi, 1 G1, 1 for all i > 1 and if i, k are primes then Gi, 1 Gk, 1 if and only if i = k.
(Received October 30 1995)