Critical groups of graphs with dihedral actions

In this paper we consider the critical group of finite connected graphs which admit harmonic actions by the dihedral group DnDn. In particular, we show that if the orbits of the DnDn-action all have either nn or 2n2n points then the critical group of such a graph can be decomposed in terms of the critical groups of the quotients of the graph by certain subgroups of the automorphism group. This is analogous to a theorem of Kani and Rosen which decomposes the Jacobians of algebraic curves with a DnDn-action.