Unimodular integer circulants associated with trinomials

The n � n circulant matrix associated with the polynomial [image removed] (with d < n) is the one with first row (a0 ? ad 0 ? 0). 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), tn-1) = �1? We give a complete answer to this question for trinomials f(t) = tm � tk � 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?Repov? generalized Fibonacci groups, thus giving an almost complete answer to a question of Bardakov and Vesnin

On congruence in ZnZ^n and the dimension of a multidimensional circulant

From a generalization to $Z^n$ of the concept of congruence we define a family of regular digraphs or graphs called multidimensional circulants, which turn to be Cayley (di)graphs of Abelian groups. This paper is mainly devoted to show the relationship between the Smith normal form for integral matrices and the dimension of such (di)graphs, that is the minimum ranks of the groups they can arise from. In particular, those 2-step multidimensional circulant which are circulants, that is Cayley (di)graphs of cyclic groups, are fully characterized. In addition, a reasoning due to Lawrence is used to prove that the cartesian product of $n$ circulants with equal number of vertice $p>2$, $p$ a prime, has dimension $n$.Peer Reviewe

Groups of Fibonacci type revisited

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 Repov?s, 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

Fibonacci type semigroups

We study "Fibonacci type" groups and semigroups. By establishing asphericity of their presentations we show that many of the groups are infinite. We combine this with Adjan graph techniques and the classification of the finite Fibonacci semigroups (in terms of the finite Fibonacci groups) to extend it to the Fibonacci type semigroups

The factors of a design matrix

AbstractLet X and Y be integral matrices of order n > 1 and suppose that these matrices satisfy the matrix equation XY = B, where B is a matrix with k in the n main diagonal positions and λ and μ in all other positions. Suppose further that k, λ, and μ are nonnegative integers and that λ occurs exactly the same number of times in each line of B and that a similar situation holds for μ. We call X and Y the factors of the design matrix B. The matrix equation described above embraces a vast category of combinatorial configurations that are characterized by square incidence matrices. We investigate the factors of design matrices and prove a duality theorem for the factors of certain “quadratic” design matrices. This result may be regarded as a strong generalization of Connor's duality theorem on symmetric group divisible designs. We conclude with a brief discussion of certain special factors of design matrices that are of particular interest to us