Redundant relators in cyclic presentations of groups

TL;DR

This paper characterizes various types of cyclic group presentations, refines their descriptions, and explores properties like the Tits alternative and largeness, providing classifications related to star graphs and generalized polygons.

Contribution

It offers a comprehensive classification of cyclic presentations, including redundant and special types, and extends understanding of their algebraic and combinatorial properties.

Findings

Tits alternative holds for groups from redundant cyclic presentations

Groups with more than two generators are large

Classifies cyclic presentations with star graphs as generalized polygons

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