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. Then we can define, on the one hand, a Poisson bracket on λ-lengths that is induced by the Poisson bracket on shear coordinates introduced by V. V. Fock in 1997 and, on the other hand, a symplectic structure Ω_WP on the set of extended shear coordinates that is induced by Penner’s symplectic structure on λ-lengths. We derive the symplectic structure Ω_WP, which turns out to be similar to Kontsevich’s symplectic structure for ψ-classes in complex analytic geometry, and demonstrate that it is indeed inverse to Fock’s Poisson structure.

中文翻译：

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Teichmüller空间上的辛结构$$ {\ mathfrak {T} _ {g，s，n}} $$ T g，s，n和簇代数

我们记得黎曼的脂肪图表描述表面Σ_克，S，N和对应的有Teichmüller空间\（{\ mathfrak【T} _ {G，S，N}} \）与S> 0的空穴和N> 0为边界双曲几何设置中的尖点。如果n> 0，则在瑟斯顿剪切坐标集和Penner的λ长度之间有一个双射。然后，我们可以定义，在一方面，泊松托架在由上剪切坐标泊松托架由VV福克引入于1997年诱导，在另一方面λ-长度，一辛结构Ω _WP上的一组扩展Penner辛结构在λ长度上引起的剪切坐标。我们推导辛结构_ΩWP，这与复杂解析几何中ψ类的Kontsevich辛结构相似，并证明它确实与Fock的Poisson结构相反。