We present a novel construction of finite groupoids whose Cayley graphs have large girth even w.r.t. a discounted distance measure that contracts arbitrarily long sequences of edges from the same colour class (sub-groupoid), and only counts transitions between colour classes (cosets). These groupoids are employed towards a generic construction method for finite hypergraphs that realise specified overlap patterns and avoid small cyclic configurations. The constructions are based on reduced products with groupoids generated by the elementary local extension steps, and can be made to preserve the symmetries of the given overlap pattern. In particular, we obtain highly symmetric, finite hypergraph coverings without short cycles. The groupoids and their application in reduced products are sufficiently generic to be applicable to other constructions that are specified in terms of local glueing operations and require global finite closure.

翻译：我们提出了一种有限广群的新构造，其Cayley图在采用折扣距离度量（该度量对同一颜色类（子广群）内任意长的边序列进行压缩，仅计算颜色类（陪集）间的转移）时仍具有大围长。这些广群被用于开发一种通用构造方法，以生成实现特定重叠模式并避免小循环构型的有限超图。该构造基于通过基本局部扩展步骤生成的广群的约化积，并可保留给定重叠模式的对称性。特别地，我们得到了无短环的高度对称有限超图覆盖。这些广群及其在约化积中的应用具有充分通用性，可推广至其他需要局部粘合操作且要求全局有限闭包的构造中。