  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.
