We associate an algebra to a triangulation of a surface with a set of boundary marking points. This algebra is gentle and Gorenstein of dimension one. We also prove that is cluster-tilted if and only if it is cluster-tilted of type or , or if and only if the surface is a disc or an annulus. Moreover all cluster-tilted algebras of type or are obtained in this way.
"Gentle algebras arising from surface triangulations." Algebra Number Theory 4 (2) 201 - 229, 2010. https://doi.org/10.2140/ant.2010.4.201