The cyclically presented groups with relators $x_ix_{i+k}x_{i+l}$

2010

Int. J. Number Theory

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The n×n circulant matrix associated with the polynomial f (t) = The problem as to when such circulants are unimodular arises in the theory of cyclically presented groups and leads to the following question, previously studied by Odoni and Cremona: when is Res(f (t), t n − 1) = ±1? We give a complete answer to this question for trinomials f (t) = t m ± t k ± 1. Our main result was conjectured by the author in an earlier paper and (with two exceptions) implies the classification of the finite Cavicchioli-Hegenbarth-Repovš generalized Fibonacci groups, thus giving an almost complete answer to a question of Bardakov and Vesnin.

“…By Lemma 4.7 we have that {w 1 , w 2 , w 3 , w 4 } = {z 1 , z 2 , z 3 , z 4 }, but z 3 = 0 ∈ {w 1 , w 2 , w 3 , w 4 } which gives a contradiction.…”

Section: Lemma 46 Letmentioning

confidence: 79%

“…In the other three cases we can reduce to a polynomial of the form t m − t k + 1; moreover we may assume (n, m, k) = 1 (see Section 3). We note that Lemma 5 of [8] and Lemma 2.3 of [4] determine when R n (t m ± t k ± 1) = 0.…”

Section: Introductionmentioning

confidence: 99%

Unimodular Integer Circulants Associated With Trinomials

2010

Int. J. Number Theory

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“…(where 0 ≤ k, l < n, and subscripts are taken mod n). This family of groups was introduced in [16] and studied in depth in [21] and further in [9]. (A reduced element of a free group is positive if all of its exponents are positive we consider groups where none of (B),(C),(D) hold and, for n < 210, we assess how effective the abelianisation is as an invariant for distinguishing non-isomorphic groups.…”

Section: Introductionmentioning

confidence: 99%

An Investigation Into the Cyclically Presented Groups with Length Three Positive Relators

Mohamed

Experimental Mathematics

2019

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We continue research into the cyclically presented groups with length three positive relators. We study small cancellation conditions and SQ-universality, we obtain the Betti numbers of the groups' abelianisations, we calculate the orders of the abelianisations of some groups, and we study isomorphism classes of the groups. Through computational experiments we assess how effective the abelianisation is as an invariant for distinguishing non-isomorphic groups.

“…The C(3)-T(6) presentations G n (x 0 x m x k ) were classified in [19,Lemma 5.1]. Extending that proof to classify the special C(3)-T(6) presentations it turns out that (up to homotopy equivalence) there is precisely one, namely G 7 (x 0 x 1 x 3 ), which was identified as being a special C(3)-T(6) presentation in [26,Example 6.3].…”

Section: ) Is a C(3)-t(7) Presentation If And Only Ifmentioning

confidence: 99%

“…The groups G n (m, k) fit into the wider classes of cyclically presented groups R(r, n, k, h) of [7] and P (r, n, k, s, q) of [38]. The related groups G n (x 0 x m x k ), introduced in [10], were studied in detail in [19].…”

Section: Introductionmentioning

confidence: 99%

Groups of Fibonacci Type Revisited

Int. J. Algebra Comput.

2012

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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.