Largeness and Sq-Universality of Cyclically Presented Groups

Williams, Gerald

doi:10.1142/s021819671250035x

Cited by 9 publications

(18 citation statements)

References 36 publications

(54 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Depending on the situation we may make some of the following assumptions to simplify the analysis. By Lemma 14 of [38] the groups P (r, n, k, s, q) ∼ = P (s, n, n − k + 2, r, q), so we may assume that r ≥ s. Also as described in [18] (see also, [ [38], Corollary 3]) the group P (r, n, k, s, q) decomposes as a free product of (n, q, k −1) copies of P (r, N, K, s, Q), where N = n/(n, q, k −1), Q = q/(n, q, k − 1) and K − 1 ≡ (k − 1)/(n, q, k − 1) mod N. In particular P (r, n, k, s, q) is perfect or trivial if and only if P (r, N, K, s, Q) is perfect or trivial respectively. Hence when suitable we may assume that (n, q, k − 1) = 1, which is the same as saying that P (r, n, k, s, q) is irreducible.…”

Section: Introductionmentioning

confidence: 92%

“…Cyclically presented groups have been an object of various investigations not just from the algebraic point of view as in [3,19,1,14,38,10,13] for example, but also from topological perspectives due to their connections with the topology of closed orientable 3-manifolds (see for example, [36,9,8,35,12,7,26]). In this article we take the former point of view.…”

Section: Introductionmentioning

confidence: 99%

“…As the name implies the group P (r, n, k, s, q) was first considered in [33] by Prishchepov; and later by others (see for example [11,36,38]). For particular choices of parameters these groups are known: Conway's Fibonacci groups F (2, n) = P (2, n, 3, 1, 1) of [15], the Cavicchioli-Hegenbarth-Repovš groups G n (m, k) = P (2, n, k + 1, 1, m) of [9,28] and more (see [5,3,35,23,4,6]).…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Perfect Prishchepov groups

Chinyere,

Bainson

2020

Preprint

View full text Add to dashboard Cite

We study cyclically presented groups of type F depending on five non-negative integer parameters n, k, q, r, s with the aim of determining when these groups are perfect. It turns out that to do so it is enough to consider the Prishchepov groups P (r, n, k, s, q), so we classify when these groups are perfect modulo some conjecture. In particular, we obtain a classification, in terms of these parameters, of the Campbell and Robertson's Fibonaccitype groups H(r, n, s) = P (r, n, r + 1, s, 1) which are perfect or trivial, thereby proving a conjecture of Williams, and yielding a complete classification of the groups H(r, n, s) that are connected Labelled Oriented Graph groups.

show abstract

Section: Introductionmentioning

confidence: 92%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Perfect Prishchepov groups

Chinyere,

Bainson

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…The Tits alternative has been considered for various classes of cyclically presented groups [17,7,22,4,11,28]. In this section we investigate the Tits alternative for the class of groups defined by redundant cyclic presentations.…”

Section: The Tits Alternativementioning

confidence: 99%

Redundant relators in cyclic presentations of groups

Chinyere¹,

Williams²

2021

Preprint

Self Cite

View full text Add to dashboard Cite

A cyclic presentation of a group is a presentation with an equal number of generators and relators that admits a particular cyclic symmetry. We characterise the orientable, non-orientable, and redundant cyclic presentations and obtain concise refinements of these presentations. We show that the Tits alternative holds for the class of groups defined by redundant cyclic presentations and that if the number of generators of the cyclic presentation is greater than two then the corresponding group is large. Generalizing and extending earlier results of the authors we describe the star graphs of orientable and non-orientable cyclic presentations and classify the cyclic presentations whose star graph components are pairwise isomorphic incidence graphs of generalized polygons, thus classifying the so-called (m, k, ν)-special cyclic presentations.

show abstract

“…In the fifty years since Conway's question in [20], the systematic study of cyclically presented groups has led to the introduction and study of numerous families of cyclic presentations with combinatorial structure determined by various parameters. These include the Fibonacci groups F (2, n) = G n (x 0 x 1 x −1 2 ) [20] and a host of generalizing families including the groups F (r, n) [33], F (r, n, k) [11], R(r, n, k, h) [14], H(r, n, s) [15], F (r, n, k, s) [16] (see also, for example, [49]), P (r, n, l, s, f ) [43], [54], H(n, m) [26], G n (m, k) = G n (x 0 x m x −1 k ) [32,17,1,53], and the groups G n (x 0 x k x l ) [18,24,7].…”

Section: Introductionmentioning

confidence: 99%

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

Bogley

Williams

2017

Journal of Algebra

Self Cite

View full text Add to dashboard Cite

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 .

show abstract

Largeness and Sq-Universality of Cyclically Presented Groups

Cited by 9 publications

References 36 publications

Perfect Prishchepov groups

Perfect Prishchepov groups

Redundant relators in cyclic presentations of groups

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

Contact Info

Product

Resources

About