Parameterizing Hitchin components
Parameterizing Hitchin components
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参数化 Hitchin 组件
DOI：
10.1215/0012794-2838654
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发表时间：
2012
影响因子：
2.5
通讯作者：
G. Dreyer
中科院分区：
文献类型：
--
作者：
F. Bonahon;G. Dreyer
我们构造了闭曲面S的PSL_n(R)-特征变量R_{PSL_n(R)}(S)的Hitchin分量Hit_n(S)的几何实解析参数化，该方法是显式的和建设性的。实质上，我们的参数化是对封闭曲面的Teichmueller空间的Thurston剪切坐标的扩展，结合了穿孔曲面的正框架局部系统的模空间的Fock-Goncharov坐标。更准确地说，给定S中具有有限多叶的最大测地层积\ λ，我们为Hitchin分量的元素引入两种不变量：与\ λ的每个叶相关的剪切不变量；以及与补S-\ λ的每个分量相关联的三角形不变量。我们描述了这些不变量所满足的恒等式和关系，并使用得到的坐标来参数化Hitchin分量。
We construct a geometric, real analytic parametrization of the Hitchin component Hit_n(S) of the PSL_n(R)-character variety R_{PSL_n(R)}(S) of a closed surface S. The approach is explicit and constructive. In essence, our parametrization is an extension of Thurston's shear coordinates for the Teichmueller space of a closed surface, combined with Fock-Goncharov's coordinates for the moduli space of positive framed local systems of a punctured surface. More precisely, given a maximal geodesic lamination \lambda in S with finitely many leaves, we introduce two types of invariants for elements of the Hitchin component: shear invariants associated with each leaf of \lambda; and triangle invariants associated with each component of the complement S-\lambda. We describe identities and relations satisfied by these invariants, and use the resulting coordinates to parametrize the Hitchin component.