Twisted Alexander ideals and the isomorphism problem for a family of parafree groups

In 1969, Baumslag introduced a family of parafree groups G(i, j) which share many properties with the free group of rank 2. The isomorphism problem for the family G(i, 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 G(i, j) associated with non-abelian representations into SL(2, Z(2)). Using the twisted Alexander ideals, we prove that several pairs of groups among G(i, j) are not isomorphic. As a consequence, we solve the isomorphism problem for sub-families containing infinitely many groups G(i, j).