Fenchel--Nielsen coordinates and Goldman brackets

TL;DR

This paper demonstrates how shear coordinates' Poisson brackets induce Fenchel--Nielsen brackets on Riemann surface parameters, extending the framework to surfaces with holes and bordered cusps.

Contribution

It explicitly connects shear coordinate Poisson brackets with Fenchel--Nielsen brackets and generalizes these structures to more complex Riemann surfaces.

Findings

Poisson bracket on shear coordinates induces Fenchel--Nielsen bracket

Extension of structures to surfaces with holes and bordered cusps

Explicit formulas for generalized parameters

Abstract

We explicitly show that the Poisson bracket on the set of shear coordinates introduced by V.V. Fock in 1997 induces the Fenchel--Nielsen bracket on the set of gluing parameters (length and twist parameters) for pairs of pants decomposition for Riemann surfaces with holes $\Sigma_{g,s}$. We generalize these structures to the case of Riemann surfaces $\Sigma_{g,s,n}$ with holes and bordered cusps.