Acyclicity in finite groups and groupoids
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We expound a simple construction of finite groups and groupoids whose Cayley graphs satisfy graded acyclicity requirements. Our acyclicity criteria concern cyclic patterns formed by coset-like configurations w.r.t. subsets of the generator set rather than cyclic configurations formed by individual generators. The proposed constructions correspondingly yield finite groups and groupoids whose Cayley graphs satisfy much stronger acyclicity conditions than large girth. We thus obtain generic and canonical constructions of highly homogeneous graph structures with strong acyclicity properties, which possess known applications in finite graph and hypergraph coverings that locally unfold cyclic configurations. An important new feature of the construction proposed here is that it reduces the hitherto considerably more complex construction for groupoids to a suitably adapted construction for groups, and even for groups with involutive generators, with the additional benefit of a more uniform approach across these settings.
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