Fenchel—Nielssen coordinates and the Goldman bracket

We show that the Poisson bracket on shear coordinates determining ideal-triangle decompositions of Riemann surfaces with holes generates, via the Goldman bracket on the set of geodesic functions and arcs, Poisson structure on the length-twist coordinates by Fenchel and Nielssen corresponding to pair-of-pants decompositions of Riemann surfaces. We give a geometric interpretation of normalized twist coordinates enjoying the canonical Poisson brackets and discuss generalizations of this construction on the case of surfaces with holes and marked points on the boundaries of the holes (boundary cusps).

References

1. L. O. Chekhov, “Koordinaty Fenkhelya–Nilsena i skobki Goldmana”, UMN, 75:5(455) (2020), 153–190      
2. L. Chekhov, M. Shapiro, Darboux coordinates for symplectic groupoid and cluster algebras, arXiv: 2003.07499

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