Highly Acyclic Groups, Hypergraph Covers and the Guarded Fragment

We construct finite groups 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, and only counts transitions between colour classes. These groups are shown to be useful in the construction of finite bisimilar hypergraph covers that avoid any small cyclic configurations. We present two applications to the finite model theory of the guarded fragment: a strengthening of the known finite model property for GF and the characterisation of GF as the guarded bisimulation invariant fragment of FO in the sense of finite model theory.