Cluster Categories of Type A∞ and Triangulations of the Infinite Strip

We call a 2-Calabi-Yau triangulated category a cluster category if its cluster-tilting subcategories form a cluster structure as defined in [4]. In this paper, we show that the canonical orbit category of the bounded derived category of finite dimensional representations of a quiver without infinite paths of type A∞ or A∞ is a cluster category. Moreover, for a cluster category of type A∞, we give a geometrical description of its cluster structure in terms of triangulations of an infinite strip with marked points in the plane.