Certain cluster algebras arise from ideal triangulations of hyperbolic Riemann surfaces. The combinatorics behind their mutations can be understood in terms of "flips" in the triangulation, and the cluster variable themselves can be interpreted in terms of Shear coordinates.

Is there a similar picture for Fenchel-Nielsen coordinates? Is there a cluster algebra associated to pants decompositions of an hyperbolic Riemann surface and is there an interpretation of its variables in terms of hyperbolic lengths of the geodesics of the pants decomposition?


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