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Twisted Alexander ideals and the isomorphism problem for a family of parafree groups
Proceedings of the Edinburgh Mathematical Society ( IF 0.7 ) Pub Date : 2020-06-22 , DOI: 10.1017/s0013091520000164 Do Viet Hung , Vu The Khoi
Proceedings of the Edinburgh Mathematical Society ( IF 0.7 ) Pub Date : 2020-06-22 , DOI: 10.1017/s0013091520000164 Do Viet Hung , Vu The Khoi
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 $SL(2,{\mathbb Z}_2)$ . 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 .
中文翻译:
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 .
更新日期:2020-06-22
中文翻译:
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