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 concise 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.

由表示定义的组，其中相应星图的组件是广义多边形的入射图，因为它们是小的抵消组，通过 Edjvet 和 Vdovina 的结果，它们是具有广义多边形作为链接的多面体的基本组，并且所以对欧几里得或双曲线建筑物采取行动；在双曲线情况下，这些组是 SQ 通用的。组的循环表示是具有相同数量的生成器和关系器的表示，它们承认特定的循环对称性。我们获得了简洁循环表示的分类，其中相应星图的组件是广义多边形。分类揭示了连通星图和不连通星图都是可能的，并且只有广义三角形（即 投影平面的关联图）和规则的完全二分图作为组件出现。我们列出了欧几里得案例中出现的表示，并表明最多有两个相应的组不是 SQ-universal（其中一个不是 SQ-universal，另一个是未解决的）。我们获得的结果表明，许多 SQ 通用组很大。