We associate an algebra A( ) to a triangulation of a surface S with a set of boundary marking points. This algebra A( ) is gentle and Gorenstein of dimension one. We also prove that A( ) is cluster-tilted if and only if it is cluster-tilted of type ‫ށ‬ or‫,ށ‬ or if and only if the surface S is a disc or an annulus. Moreover all cluster-tilted algebras of type ‫ށ‬ or‫ށ‬ are obtained in this way.

We apply our previous work on cluster characters for Hom-infinite cluster categories to the theory of cluster algebras. We give a new proof of Conjectures 5.4, 6.13, 7.2, 7.10 and 7.12 of Fomin and Zelevinsky's Cluster algebras IV [Compositio Math. 143 (2007), 112-164] for skew-symmetric cluster algebras. We also construct an explicit bijection sending certain objects of the cluster category to the decorated representations of Derksen, Weyman and Zelevinsky, and show that it is compatible with mutations in both settings. Using this map, we give a categorical interpretation of the E-invariant and show that an arbitrary decorated representation with vanishing E-invariant is characterized by its g-vector. Finally, we obtain a substitution formula for cluster characters of not necessarily rigid objects.