2010
DOI: 10.1142/s1793042110003289
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Unimodular Integer Circulants Associated With Trinomials

Abstract: 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-Hegenba… Show more

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Cited by 12 publications
(17 citation statements)
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“…In connection with this and with Question 5 of [1] we note that this behaviour cannot occur for the groups H n .m; k/ (of the introduction) when they are finite (by [14], [15]), and that there are no recorded examples of it when they are infinite.…”
Section: Preliminariesmentioning
confidence: 64%
“…In connection with this and with Question 5 of [1] we note that this behaviour cannot occur for the groups H n .m; k/ (of the introduction) when they are finite (by [14], [15]), and that there are no recorded examples of it when they are infinite.…”
Section: Preliminariesmentioning
confidence: 64%
“…If r ≥ 3 then R(r, n, k, h) and T (r, n, k, h) are infinite by Theorem A, so we may assume r = 2. If α ≡ 0 and β ≡ 0 mod n then R(2, n, k, h), and hence T (2, n, k, h), is infinite by [17], [18]. If α ≡ 0 or β ≡ 0 mod n then T (r, n, k, h) is the semigroup free product of (n, h) copies of T (2, n, k, 1) = S(2, n, k) which, by [16,Theorem 4], is the union of n trivial ideals, and hence T (r, n, k, h) is infinite.…”
Section: Proof Of Theorem Bmentioning
confidence: 99%
“…Except for two groups, this was provided in [13], [17], [18], [10]. The unresolved groups are the Gilbert-Howie groups ( [13]) H(9, 4) = R (2,9,6,4) and H(9, 7) = R(2, 9, 3, 7).…”
Section: Proof Of Theorem Bmentioning
confidence: 99%
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“…, x n−1 | x i x i+m = x i+k (0 ≤ i ≤ n − 1) (0 ≤ m, k ≤ n − 1) were introduced in [29]. Following the appearance of the groups G n (m, 1) in [21] in connection with Labelled Oriented Graph groups and the independent re-introduction of the groups G n (m, k) in [8] they have enjoyed renewed interest over the last decade [1], [9], [27], [46], [47], [48]. The groups G n (m, k) generalize various groups that have previously been studied: G n (1, 2) are Conway's Fibonacci groups F (2, n) of [12], the groups G n (2, 1) are the Sieradski groups S(2, n) of [41], and the groups G n (m, 1) are the Gilbert-Howie groups H(n, m) of [21].…”
Section: Introductionmentioning
confidence: 99%