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In the context of representation theory of finite dimensional algebras,string algebras have been extensively studied and most aspects of theirrepresentation theory are well-understood. One exception to this is theclassification of extensions between indecomposable modules. In this paper weexplicitly describe such extensions for a class of string algebras, namelygentle algebras associated to surface triangulations. These algebras arise asJacobian algebras of unpunctured surfaces. We relate the extension spaces ofindecomposable modules to crossings of arcs in the surface and give explicitbases of the extension spaces for indecomposable modules in almost all cases.We show that the dimensions of these extension spaces are given in terms ofcrossing arcs in the surface.  Our approach is new and consists of interpreting snake graphs asindecomposable modules. In order to show that our basis is a spanning set, weneed to work in the associated cluster category where we explicitly calculatethe middle terms of extensions and give bases of their extension spaces. Wenote that not all extensions in the cluster category give rise to extensionsfor the Jacobian algebra.

Extensions in Jacobian Algebras and Cluster Categories of Marked Surfaces