Redundant relators in cyclic presentations of groups

Abstract

Abstract 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 show that if the number of generators of the cyclic presentation is greater than two, then the corresponding group is large. Generalising 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 generalised polygons, thus classifying the so-called ( m , k , ν ) (m,k,\nu) -special cyclic presentations.