One-relator quotients of free products of cyclic groups

Edjvet, Martin; Juhász, Arye

doi:10.1080/00927870008826867

Cited by 5 publications

(14 citation statements)

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“…Freiheitssatz theorems were obtained by Shwartz for Case 1 in [36], for Case 2 in [37], and for Case 3 in [38]; all of these results are contained in [35]. Case 0 was considered in [17], [18] and our arguments below may be applied to this case; however, since these results are more intricate we limit ourselves to applying the results of Cases …”

Section: Freiheitssatz Methods For Prischepov Groupsmentioning

confidence: 99%

Section: Proofmentioning

confidence: 99%

“…A one-relator product G = (H * K)/ << R >> (where << R >> denotes the normal closure of R in H * K) is said to satisfy the Freiheitssatz if the natural homomorphisms H → G, K → G are both embeddings. The Freiheitssatz for one-relator products has been considered in many papers -see [17], [18], [26], [38] and the references therein. Setting…”

Section: Amalgamated Free Products and The Freiheitssatzmentioning

confidence: 99%

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Largeness and Sq-Universality of Cyclically Presented Groups

Williams

2012

Int. J. Algebra Comput.

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Largeness, SQ-universality, and the existence of free subgroups of rank 2 are measures of the complexity of a finitely presented group. We obtain conditions under which a cyclically presented group possesses one or more of these properties. We apply our results to a class of groups introduced by Prishchepov which contain, amongst others, the various generalizations of Fibonacci groups introduced by Campbell and Robertson. Using the techniques developed we give a new, purely group-theoretic, proof of the (almost complete) classification of the finite Cavicchioli-HegenbarthRepovš groups.

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Section: Freiheitssatz Methods For Prischepov Groupsmentioning

confidence: 99%

Section: Proofmentioning

confidence: 99%

Section: Amalgamated Free Products and The Freiheitssatzmentioning

confidence: 99%

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Largeness and Sq-Universality of Cyclically Presented Groups

Williams

2012

Int. J. Algebra Comput.

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“…Since H ∼ = Z, if the Freiheitssatz holds in this setting we have that E n (m, k) (and hence G n (m, k)) is infinite. Now R is of the form R = abcd (a, c ∈ H, b, d ∈ K), and the Freiheitssatz for one-relator products where the relator takes this form was studied in [17], [18], [39], [40] and other papers by the same authors. In our proof we will require results from those papers including the following, which we reproduce here as the preprint [39] remains unpublished.…”

Section: The Finite Fibonacci Groups and Sieradski Groupsmentioning

confidence: 99%

Groups of Fibonacci Type Revisited

Williams

2012

Int. J. Algebra Comput.

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This article concerns a class of groups of Fibonacci type introduced by Johnson and Mawdesley that includes Conway's Fibonacci groups, the Sieradski groups, and the Gilbert-Howie groups. This class of groups provides an interesting focus for developing the theory of cyclically presented groups and, following questions by Bardakov and Vesnin and by Cavicchioli, Hegenbarth, and Repovš, they have enjoyed renewed interest in recent years. We survey results concerning their algebraic properties, such as isomorphisms within the class, the classification of the finite groups, small cancellation properties, abelianizations, asphericity, connections with Labelled Oriented Graph groups, and the semigroups of Fibonacci type. Further, we present a new method of proving the classification of the finite groups that deals with all but three groups.

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“…In another direction, recent work of Edjvet, Juhász and Shwartz [4,7,8,17,18,19] shows that the Freiheitssatz often holds for short relations. Now a one-relator product of (finite) cyclic groups, in which the relator is a power, is called a generalized triangle group.…”

Section: Introductionmentioning

confidence: 99%

One-Relator Products Induced from Generalized Triangle Groups

Howie

Shwartz

2004

Communications in Algebra

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Abstract. We study the quotient of a free product of groups by the normal closure of a word that is contained in a certain type of 2-generator subgroup, and hence can be expressed in the form of a pushout involving a generalized triangle group. Using the theory of generalized triangle groups, we extend the range of conditions under which a number of results for one-relator products of groups, such as the Freiheitssatz and the solubility of the word problem, are known to hold.

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One-relator quotients of free products of cyclic groups

Cited by 5 publications

References 5 publications

Largeness and Sq-Universality of Cyclically Presented Groups

Largeness and Sq-Universality of Cyclically Presented Groups

Groups of Fibonacci Type Revisited

One-Relator Products Induced from Generalized Triangle Groups

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