On shift dynamics for cyclically presented groups

Bogley, William A.

doi:10.1016/j.jalgebra.2014.07.009

Cited by 17 publications

(36 citation statements)

References 31 publications

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“…The orientability condition in Theorem 3.4 is necessary to ensure that dipoles represent homotopically trivial elements of π 2 (L(P), K(G, 1)). See [12, Figure 1] for an example of a dipole in the non-orientable setting and see [10,Example 3.5] for additional discussion. Diagrammatic reducibility and asphericity are distinct concepts for both ordinary and relative group presentations.…”

Section: Diagrammatic Reducibilitymentioning

confidence: 99%

Aspherical Relative Presentations all Over Again

Williams¹,

Bogley²,

Edjvet³

2019

Groups St Andrews 2017 in Birmingham

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In gratitude to Stephen J. Pride for his friendship and mentorship over the years. AbstractThe concept of asphericity for relative group presentations was introduced twenty five years ago. Since then, the subject has advanced and detailed asphericity classifications have been obtained for various families of one-relator relative presentations. Through this work the definition of asphericity has evolved and new applications have emerged.In this article we bring together key results on relative asphericity, update them, and exhibit them under a single set of definitions and terminology. We describe consequences of asphericity and present techniques for proving asphericity and for proving non-asphericity. We give a detailed survey of results concerning one-relator relative presentations where the relator has free product length four.

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Section: Diagrammatic Reducibilitymentioning

confidence: 99%

Aspherical Relative Presentations all Over Again

Williams¹,

Bogley²,

Edjvet³

2019

Groups St Andrews 2017 in Birmingham

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“…The various combinations of conditions represented in the lefthand column of Table 2 are analyzed individually in Sections 6-9. As in [3,8,14], we depend on previous work concerning asphericity of relative presentations, in this case [2]. As is typical in the study of relative asphericity, most of the presentations encountered in [2] are aspherical, with the nonaspherical cases falling into well-defined infinite families (see Theorem 9.2.1), along with a handful of isolated nonaspherical cases.…”

Section: Cyclically Presented Groups and Shift Dynamicsmentioning

confidence: 99%

“…In type (I5), G is metacyclic and non-nilpotent of order 220. The types (I6 ′ ) and (I6 ′′ ) lead to nonisomorphic nonsolvable groups G of order 4 088 448 = 2 7 · 3 3 · 7 · 13 2 , each containing the simple group PSL (3,3).…”

Section: Conditionmentioning

confidence: 99%

“…Asphericity and freeness are related by a general result from [3]. A cyclic presentation P n (w) is orientable if w is not a cyclic permutation of the inverse of any of its shifts.…”

Section: Conditionmentioning

confidence: 99%

“…In [3,Theorems B,C] it was shown that if w = x 0 x k x l is a positive word of length three, then the relationships between finiteness, asphericity, and shift dynamics are sharp:…”

Section: Conditionmentioning

confidence: 99%

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Cyclically presented groups with length four positive relators

Bogley

Parker

2018

Journal of Group Theory

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Coherence, subgroup separability, and metacyclic structures for a class of cyclically presented groups

Bogley

Williams

2017

Journal of Algebra

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We study a class M of cyclically presented groups that includes both finite and infinite groups and is defined by a certain combinatorial condition on the defining relations. This class includes many finite metacyclic generalized Fibonacci groups that have been previously identified in the literature. By analysing their shift extensions we show that the groups in the class M are are coherent, subgroup separable, satisfy the Tits alternative, possess finite index subgroups of geometric dimension at most two, and that their finite subgroups are all metacyclic. Many of the groups in M are virtually free, some are free products of metacyclic groups and free groups, and some have geometric dimension two. We classify the finite groups that occur in M, giving extensive details about the metacyclic structures that occur, and we use this to prove an earlier conjecture concerning cyclically presented groups in which the relators are positive words of length three. We show that any finite group in the class M that has fixed point free shift automorphism must be cyclic.(3) onto the cyclic subgroup of order n, generated by t, that is given by ν f (t) = t and ν f (y) = t f .Proof. Working in the free group F with basis x 0 , . . . , x n−1 , we are given that G ∼ = G n (w) where w is the product of two f -blocks, so that w = θ a1 (Λ(r 1 , f )) δ · θ a2 (Λ(r 2 , f )) ǫ where r 1 , r 2 ≥ 0 and δ, ǫ = ±1. Up to cyclic permutation and inversion, we may assume that a 1 = 0 and δ = +1. Setting a = a 2 , the word w has the form w = Λ(r 1 , f ) · θ a (Λ(r 2 , f )) ǫ . Introducing x, y, t and setting x i = t i xt −i and y = xt f , the word w transforms to W = y r1 t −r1f · t a (y r2 t −r2f ) ǫ t −a and the shift extension G n (w) ⋊ θ Z n admits a presentation y, t | t n , W . If ǫ = +1, then the first claim follows by setting r = r 1 , s = −r 2 , A = a − r 1 f , and B = a + r 2 f , while if ǫ = −1, we set r = r 1 , s = r 2 , A = a + (r 2 − r 1 )f , and B = a. Conversely, given the retraction ν f : E → Z n , a standard Reidemeister-Schreier process (see e.g. [7, Theorem 2.3]) shows that ker ν f ∼ = G n (w) ∼ = G where w is the product of two f -blocks.Example 2 (Non-isomorphic cyclically presented groups sharing the same extension). The generalized Fibonacci group F (3, 6) is G 6 (x 0 x 1 x 2 x −1 3 ) and has extension E = F (3, 6) ⋊ θ Z 6 ∼ = t, x | t 6 , xtxtxtx −1 t −3 y=xt ∼ = t, y | t 6 , y 3 ty −1 t −3 .

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On shift dynamics for cyclically presented groups

Cited by 17 publications

References 31 publications

Aspherical Relative Presentations all Over Again

Aspherical Relative Presentations all Over Again

Cyclically presented groups with length four positive relators

Coherence, subgroup separability, and metacyclic structures for a class of cyclically presented groups

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