Asphericity of symmetric presentations

Spaggiari, F.

doi:10.5565/publmat_50106_07

Cited by 6 publications

(6 citation statements)

References 14 publications

(18 reference statements)

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“…• N. D. Gilbert and J. Howie established a connection between asphericity of relative presentations (in the sense of [3]) and topological asphericity of two-dimensional cellular models of cyclic presentations in [15, Lemma 3.1]. The same argument was adapted to other classes of presentations in [2,5,36]. Edjvet and Williams established a complete characterization of topological asphericity for two-dimensional cellular models of the cyclic presentations P n (k, l) in [12,Corollary 4.3], correcting an error in [5,Theorem 4.3].…”

Section: Proof Of Theorem Amentioning

confidence: 99%

On shift dynamics for cyclically presented groups

Bogley

2014

Journal of Algebra

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For group presentations with cyclic symmetry, there is a connection between asphericity and the dynamics of the shift automorphism. For the class of groups G n (k, l) described by the cyclic presentations P n (k, l) = (x i : x i x i+k x i+l (i mod n)) and studied extensively by G. Williams and M. Edjvet [12], the shift acts freely on the nonidentity elements of G n (k, l) if and only if the presentation P n (k, l) is combinatorially aspherical in the sense of [6]. The shift has a nonidentity fixed point precisely when G n (k, l) is finite. MSC (2010) 20F05, 20E36.

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Section: Proof Of Theorem Amentioning

confidence: 99%

On shift dynamics for cyclically presented groups

Bogley

2014

Journal of Algebra

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“…The asphericity of cyclically presented groups and generalizations of Fibonacci groups have been studied in [1], [6], [11], [14], [16]. In this paper we consider the asphericity of presentations G n ðm; kÞ.…”

Section: Introductionmentioning

confidence: 99%

The aspherical Cavicchioli–Hegenbarth–Repovš generalized Fibonacci groups

Williams¹

2009

Journal of Group Theory

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Abstract. The Cavicchioli-Hegenbarth-Repovš generalized Fibonacci groups are defined by the presentations G n ðm; kÞ ¼ hx 1 ; . . . ; x n j x i x iþm ¼ x iþk ð1 c i c nÞi. These cyclically presented groups generalize Conway's Fibonacci groups and the Sieradski groups. Building on a theorem of Bardakov and Vesnin we classify the aspherical presentations G n ðm; kÞ. We determine when G n ðm; kÞ has infinite abelianization and provide su‰cient conditions for G n ðm; kÞ to be perfect. We conjecture that these are also necessary conditions. Combined with our asphericity theorem, a proof of this conjecture would imply a classification of the finite CavicchioliHegenbarth-Repovš groups.

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“…Theorem 5.1 is significant because it gives examples of infinite cyclically presented groups with torsion. Indeed, in many studies (for example [15,35,2,8,12,5,36]) cyclically presented groups are proved infinite by showing that they are non-trivial and that the relative presentations of their shift extensions are aspherical, and deducing (by [15,Lemma 3.1], [3, Theorem 4.1(a)]) that the cyclic presentation is topologically aspherical, and hence that the cyclically presented group is torsion-free.…”

Section: Torsion and Asphericitymentioning

confidence: 99%