Symplectic structures on Teichmüller spaces $\mathfrak T_{g,s,n}$ and cluster algebras

We recall the fat-graph description of Riemann surfaces $Σ_{g,s,n}$ and the corresponding Teichmüller spaces $\mathfrak T_{g,s,n}$ with $s>0$ holes and $n>0$ bordered cusps in the hyperbolic geometry setting. If $n>0$, we have a bijection between the set of Thurston shear coordinates and Penner's $λ$-lengths and we can induce, on the one hand, the Poisson bracket on $λ$-lengths from the Poisson bracket on shear coordinates introduced by V.V.Fock in 1997 and, on the other hand, a symplectic structure $Ω_{\text{WP}}$ on the set of extended shear coordinates from Penner's symplectic structure on $λ$-lengths. We derive $Ω_{\text{WP}}$, which turns out to be similar to the Kontsevich symplectic structure for $ψ$-classes in complex-analytic geometry, and demonstrate that it is indeed inverse to the Fock Poisson structure.

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