Acyclicity in finite groups and groupoids

We expound a concise 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 just 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 support known applications in finite graph and hypergraph coverings that locally unfold cyclic configurations.
Comment: An error in (v1) invalidated a more direct reduction of the groupoidal case to the group case. The current version further corrects and expands some of the core technical arguments, and introduces some core concepts more systematically (partly in dedicated new sections). See acknowledgements for a more detailed account; (v5/6) contain minor corrections and updates