Algebras from Orbifolds

We discuss two algebras associated to triangulated unpunctured orbifolds with all orbifold points of order three - a gentle algebra and a generalized cluster algebra, in the sense of Chekhov and Shapiro. To each algebra, we associate a map which can be seen as taking arcs on the orbifold to Laurent polynomials. The first map was defined by Caldero and Chapoton; the second is the snake graph map, defined for surfaces by Musiker-Schiffler-Williams and for orbifolds by B.-Kelley. We show that the outputs of these two maps agree. This talk is based on joint work with Yadira Valdivieso.