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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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  • $\begingroup$ I would personally love such an interpretation. The Fenchel-Nielsen coordinates have an algebraic structure that looks remarkably similar to the cluster algebra structure, but I don't know how to make it look as neat. It's possible the right point of view is to make a "triangulation" with spun ideal triangles (spinning along the pants curves), $\endgroup$ Commented Apr 29, 2021 at 21:33
  • $\begingroup$ A related question: there is a map (the shear map) from the decorated teichmuller space to a subspace of the teichmuller space of a surface with holes. The map lands in the subspace of the latter where holes have length 0 (so that they are effectively punctures). Is there a generalization of this map whose image covers all of the target Teichmuller space? $\endgroup$ Commented Jun 3, 2021 at 12:17
  • $\begingroup$ that's really a separate question, and not really precise enough for me to answer as stated. $\endgroup$ Commented Jun 4, 2021 at 17:01
  • $\begingroup$ Apologise, I'll turn it into an actual question and be more precise. $\endgroup$ Commented Jun 4, 2021 at 17:31
  • $\begingroup$ Here it is: mathoverflow.net/q/394542/48526 I hope I have managed to be more clear now. $\endgroup$ Commented Jun 4, 2021 at 17:45

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