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.

中文翻译：

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扭曲的 Alexander 理想和 Parafree 群族的同构问题

1969 年，Baumslag 引入了一个 parafree 组家族G一世,j与秩为 2 的自由群共享许多性质。族的同构问题G一世,j已知困难；到目前为止，已经发现了一些小的部分结果。在本文中，我们计算了群的扭曲亚历山大理想G一世,j与非阿贝尔表示相关联$SL(2,{\mathbb Z}_2)$. 使用扭曲的亚历山大理想，我们证明了G一世,j不是同构的。因此，我们解决了包含无限多组的子族的同构问题G一世,j.