In this paper we use the Alexander ideals of groups to solve the isomorphism problem for the Baumslag–Solitar groups and a family of parafree groups introduced by Baumslag and Cleary.

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In 1969, Baumslag introduced a family of parafree groups Gi,j which share many properties with the free group of rank 2. The isomorphism problem for the family Gi,j is known to be difficult; a few small partial results have been found so far. In this paper, we compute the twisted Alexander ideals of the groups Gi,j associated with non-abelian representations into $SL(2,{\mathbb Z}_2)$ . Using the twisted Alexander ideals, we prove that several pairs of groups among Gi,j are not isomorphic. As a consequence, we solve the isomorphism problem for sub-families containing infinitely many groups Gi,j.

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