A geometric description of m-cluster categories

We show that the m-cluster category of type An−1 is equivalent to a certain geometrically defined category of diagonals of a regular nm + 2-gon. This generalises a result of Caldero, Chapoton and Schiffler for m=1. The approach uses the theory of translation quivers and their corresponding mesh categories. We also introduce the notion of the m-th power of a translation quiver and show how it can be used to realise the m-cluster category in terms of the cluster category.