Groups with the Same lower Central Sequence as a Relatively Free Group. II. Properties

Baumslag, Gilbert

doi:10.2307/1995369

Cited by 19 publications

(12 citation statements)

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“…Then, by Propositions 4.1-4.3, the twisted Alexander ideals of G i,j are I 1 i,j = I 2 i,j = L and I 3 i,j = (1 + y d + y 2d ). As 3|(i + j ), 3 | j and l is odd, by Proposition 4.3 we have I 3 i ,j = ((1 + y d + y 2d )/(1 + y + y 2 )). Note that as d > 1, I 3 i ,j = L. Since G i,j ∼ = G i ,j , two of the twisted Alexander ideals of G i ,j must coincide with L. For this to happen, we must have I 1 i ,j = I 2 i ,j = L. Since the automorphism ϕ must map a non-trivial ideal to a Since the automorphism ϕ must map a non-trivial ideal to a non-trivial one, we get ϕ(I 3 i,j ) = I 3 i ,j .…”

Section: Applications To the Isomorphism Problemmentioning

confidence: 93%

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

Hung

2020

Proceedings of the Edinburgh Mathematical Society

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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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Section: Applications To the Isomorphism Problemmentioning

confidence: 93%

“…In this paper, we study a family of parafree groups G i,j which was introduced by Baumslag in [2,3]:…”

Section: Introductionmentioning

confidence: 99%

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

Hung

2020

Proceedings of the Edinburgh Mathematical Society

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“…The isomorphism problem is one of the many open problems about these groups. Now G. Baumslag [2] has proved that the groups…”

Section: Parafree Groupsmentioning

confidence: 99%

Experimenting and computing with infinite groups

Baumslag¹,

Miller²

1997

Groups and Computation II

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We describe an experimental approach to studying infinite groups using a software package called Magnus being developed by the New York Group Theory Cooperative. This approach emphasises infinite groups and partial and experimental computation. These computations are frequently inconclusive and may only occasionally succeed. Such experimentation can guide theoretical development and lead to new and interesting questions.

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“…These Lie algebras arose from Baumslag's works about parafree groups. In [2][3][4], Baumslag has introduced parafree groups and he obtained some interesting results about these groups. In his doctoral dissertation [1], Baur has defined parafree Lie algebras as in the group case and he proved the existence of a nonfree parafree Lie algebra [5].…”

Section: Introductionmentioning

confidence: 99%