Aspherical word labeled oriented graphs and cyclically presented groups

Harlander, Jens; Rosebrock, Stephan

doi:10.1142/s021821651550025x

Cited by 5 publications

(3 citation statements)

References 11 publications

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“…We remark that what we call a labeled oriented graph is elsewhere called a weakly labeled oriented graph or word-labeled oriented graph. See Howie [12] and Harlander and Rosebrock [10].…”

Section: Groups Defined By Graphsmentioning

confidence: 99%

Ribbon 2–knot groups of Coxeter type

Harlander,

Rosebrock

2023

Algebr. Geom. Topol.

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Wirtinger presentations of deficiency 1 appear in the context of knots, long virtual knots, and ribbon 2-knots. They are encoded by labeled oriented trees and, for that reason, are also called LOT presentations. These presentations are a well known and important testing ground for the validity (or failure) of Whitehead's asphericity conjecture. We define LOTs of Coxeter type and show that for every given n there exists a prime LOT of Coxeter type with group of rank n. We also show that label separated Coxeter LOTs are aspherical. 20F05, 20F06, 20F65; 57K20, 57K45 Dedicated to the memory of Stephen Pride 1 IntroductionWirtinger presentations of deficiency 1 appear in the context of knots, long virtual knots, and ribbon 2-knots; see Harlander and Rosebrock [9]. They are encoded by labeled oriented trees and, for that reason, are also called LOT presentations. Adding a generator to the set of relators in a Wirtinger presentation P gives a balanced presentation of the trivial group. Thus the associated 2-complex K.P / is a subcomplex of an aspherical (in fact contractible) 2-complex. Wirtinger presentations are a well-known and important testing ground for the validity (or failure) of Whitehead's asphericity conjecture, which states that a subcomplex of an aspherical 2-complex is aspherical. For more on the Whitehead conjecture see Bogley [3], Berrick and Hillman [1] and Rosebrock [18].If P is a Wirtinger presentation and the group G.P / defined by P is a 1-relator group, then G.P / admits a 2-generator 1-relator presentation P 0 and the corresponding 2complex K.P 0 / is aspherical. Since K.P 0 / and K.P / have the same Euler characteristic and the same fundamental group, it follows (using Schanuel's lemma and Kaplansky's theorem, which states that finitely generated free ZG-modules are Hopfian) that K.P /

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“…We remark that what we call a labeled oriented graph is elsewhere called a weakly labeled oriented graph or word-labeled oriented graph. See Howie [12] and Harlander and Rosebrock [10].…”

Section: Groups Defined By Graphsmentioning

confidence: 99%

Ribbon 2–knot groups of Coxeter type

Harlander,

Rosebrock

2023

Algebr. Geom. Topol.

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“…An almost complete classification of groups H(r, n, s) that are connected LOG groups was given in [34]; Chinyere and Bainson classify the perfect groups H(r, n, s) [9], completing the connected LOG groups classification. Asphericity of certain cyclic presentations of the form P n (x 0 wx −1 1 w −1 ) that are (connected) Word Labelled Oriented Graph presentations (or Wirtinger presentations) is established in [13,Section 3]. In this article we investigate when cyclically presented groups are LOG groups or connected LOG groups.…”

Section: Introductionmentioning

confidence: 99%

Cyclically presented groups as Labelled Oriented Graph groups

Noferini¹,

Williams²

2021

Preprint

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We use results concerning the Smith forms of circulant matrices to identify when cyclically presented groups have free abelianisation and so can be Labelled Oriented Graph (LOG) groups. We generalize a theorem of Odoni and Cremona to show that for a fixed defining word, whose corresponding representer polynomial has an irreducible factor that is not cyclotomic and not equal to ±t, there are at most finitely many n for which the corresponding n-generator cyclically presented group has free abelianisation. We classify when Campbell and Robertson's generalized Fibonacci groups H(r, n, s) are LOG groups and when the Sieradski groups are LOG groups. We prove that amongst Johnson and Mawdesley's groups of Fibonacci type, the only ones that can be LOG groups are Gilbert-Howie groups H(n, m). We conjecture that if a Gilbert-Howie group is a LOG group, then it is a Sieradski group, and prove this in certain cases (in particular, for fixed m, the conjecture can only be false for finitely many n). We obtain necessary conditions for a cyclically presented group to be a connected LOG group in terms of the representer polynomial and apply them to the Prishchepov groups.

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“…Connections between HNN extensions of cyclically presented groups and LOG groups have been investigated in [14], [31], [19]. Asphericity of certain cyclic presentations that are (connected) word labelled oriented graph (WLOG) presentations is established in [17,Section 3]. In this paper we investigate a particular family of cyclically presented groups and aim to classify when they are connected LOG groups or when they are knot groups.…”

Section: Introductionmentioning

confidence: 99%

Generalized Fibonacci groupsH(r,n,s) that are connected labelled oriented graph groups

Williams

2018

Journal of Group Theory

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The class of connected LOG (Labelled Oriented Graph) groups coincides with the class of fundamental groups of complements of closed, orientable 2-manifolds embedded in S 4 , and so contains all knot groups. We investigate when Campbell and Robertson's generalized Fibonacci groups H(r, n, s) are connected LOG groups. In doing so, we use the theory of circulant matrices to calculate the Betti numbers of their abelianizations. We give an almost complete classification of the groups H(r, n, s) that are connected LOG groups. All torus knot groups and the infinite cyclic group arise and we conjecture that these are the only possibilities. As a corollary we show that H(r, n, s) is a 2-generator knot group if and only if it is a torus knot group.

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Aspherical word labeled oriented graphs and cyclically presented groups

Cited by 5 publications

References 11 publications

Ribbon 2–knot groups of Coxeter type

Ribbon 2–knot groups of Coxeter type

Cyclically presented groups as Labelled Oriented Graph groups

Generalized Fibonacci groupsH(r,n,s) that are connected labelled oriented graph groups

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