Fenchel--Nielsen coordinates and Goldman brackets

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.