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 D n . In particular, we show that if the orbits of the D n -action all have either n or 2 n 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 D n -action.