Narrow Results

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 Labeled 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.

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.