Balanced Presentations of the Trivial Group

Burns, R. G.; Macedońska, O.

doi:10.1112/blms/25.6.513

Cited by 26 publications

(30 citation statements)

References 4 publications

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“…Some candidates for counterexamples, that is, for having infinite recalcitrance, are given in [2]. One formerly hopeful such candidate (not in [2]), namely the annihilating pair {yxyx~1y~1x~1, x 2 y~3} for F2, the free group on x,y, has recently been shown by computer to have finite recalcitrance after all [3]. That in fact this pair has recalcitrance 2 can be shown as follows.…”

Section: Introductionmentioning

confidence: 99%

“…For some purposes -see [2] and the remarks below -the latter transformations are more convenient than those of Andrews and Curtis, and it is these that we shall take as our "elementary moves" here.…”

Section: Introductionmentioning

confidence: 99%

“…The concept of recalcitrance has its origin in the paper [1] of Andrews and Curtis, which contains the by now well-known conjecture that in the free group F n of rank n every annihilating n-tuple has finite recalcitrance. Some candidates for counterexamples, that is, for having infinite recalcitrance, are given in [2]. One formerly hopeful such candidate (not in [2]), namely the annihilating pair {yxyx~1y~1x~1, x 2 y~3} for F2, the free group on x,y, has recently been shown by computer to have finite recalcitrance after all [3].…”

Section: Introductionmentioning

confidence: 99%

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Recalcitrance in groups

Burns

Herfort

Kam

et al. 1999

Bull. Austral. Math. Soc.

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Motivated by a well-known conjecture of Andrews and Curtis, we consider the question as to how in a given n-generator group G, a given set of n “annihilators” of G, that is, with normal closure all of G, can be transformed by standard moves into a generating n-tuple. The recalcitrance of G is defined to be the least number of elementary standard moves (”elementary M-transformations”) by means of which every annihilating n-tuple of G can be transformed into a generating n-tuple. We show that in the classes of finite and soluble groups, having zero recalcitrance is equivalent to nilpotence, and that a large class of 2-generator soluble groups has recalcitrance at most 3. Some examples and remarks are included.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Recalcitrance in groups

Burns

Herfort

Kam

et al. 1999

Bull. Austral. Math. Soc.

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“…Either of them is a composition of several transformations (AC1)-(AC3). The first transformation comes from [7], where it is called "Basic substitution principle". Applying an automorphism.…”

Section: Ac-equivalent Presentationsmentioning

confidence: 99%

“…The first two examples are probably the most established ones; see [7] for discussion. The presentation (3) is attributed to M.Wicks (by folklore), but we do not have a relevant reference.…”

Section: Introductionmentioning

confidence: 99%

On the Andrews-Curtis equivalence

Myasnikov¹,

Shpilrain²

2002

Contemporary Mathematics

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Abstract. The Andrews-Curtis conjecture claims that every balanced presentation of the trivial group can be reduced to the standard one by a sequence of "elementary transformations" which are Nielsen transformations augmented by arbitrary conjugations. It is a prevalent opinion that this conjecture is false; however, not many potential counterexamples are known. In this paper, we show that some of the previously proposed examples are actually not counterexamples. We hope that the tricks we used in constructing relevant chains of elementary transformations will be useful to those who attempt to establish the Andrews-Curtis equivalence in other situations.On the other hand, we give two rather general and simple methods for constructing balanced presentations of the trivial group; some of these presentations can be considered potential counterexamples to the Andrews-Curtis conjecture. One of the methods is based on a simple combinatorial idea of composition of group presentations, whereas the other one uses "exotic" knot diagrams of the unknot.We also consider the Andrews-Curtis equivalence in metabelian groups and reveal some interesting connections of relevant problems to well-known problems in K-theory.

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On balanced presentations of the trivial group

Ivanov

2006

Invent. math.

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Balanced Presentations of the Trivial Group

Cited by 26 publications

References 4 publications

Recalcitrance in groups

Recalcitrance in groups

On the Andrews-Curtis equivalence

On balanced presentations of the trivial group

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