Generalized polygons and star graphs of cyclic presentations of groups

TL;DR

This paper classifies cyclic group presentations with star graphs as generalized polygons, revealing their geometric structures and properties, including their actions on Euclidean or hyperbolic buildings and SQ-universality.

Contribution

It provides a complete classification of non-redundant cyclic presentations with star graphs as generalized polygons, identifying the types of graphs and analyzing their geometric and algebraic properties.

Findings

Connected and disconnected star graphs are possible.

Only generalized triangles and complete bipartite graphs arise as components.

Most groups are SQ-universal and large.

Abstract

Groups defined by presentations for which the components of the corresponding star graph are the incidence graphs of generalized polygons are of interest as they are small cancellation groups that - via results of Edjvet and Vdovina - are fundamental groups of polyhedra with the generalized polygons as links and so act on Euclidean or hyperbolic buildings; in the hyperbolic case the groups are SQ-universal. A cyclic presentation of a group is a presentation with an equal number of generators and relators that admits a particular cyclic symmetry. We obtain a classification of the non-redundant cyclic presentations where the components of the corresponding star graph are generalized polygons. The classification reveals that both connected and disconnected star graphs are possible and that only generalized triangles (i.e. incidence graphs of projective planes) and regular complete bipartite graphs arise as the components. We list the presentations that arise in the Euclidean case and show that at most two of the corresponding groups are not SQ-universal (one of which is not SQ-universal, the other is unresolved). We obtain results that show that many of the SQ-universal groups are large.