Hitchin components for orbifolds

Teichmüller空间参数化曲面上的双曲结构集。它们也可以被看作是PSL（2，R）中曲面基本群的离散和忠实表示的空间。高等泰希米勒理论试图推广泰希米勒空间的理论，用另一个李群G代替群PSL（2，R）。在这个项目中，G通常是PSL（n，R）。希钦表示是PSL（n，R）中曲面群的一种特殊表示，具有与Fuchsian表示相似的性质。这种表示的集合被称为希钦分量，它们推广了Teichmüller空间。在这个项目中，希钦分量的研究将推广到更一般的群族，即二维orbifolds的基本群。这是一个大家族的群体，包括例如2维双曲考克斯特群。PSL（2，R）的情形已经由Thurston研究过，他在orbifolds上引入了双曲结构并研究了它们的Teichmüller空间。Choi和Goldman研究了PSL（3，R）的情形。本项目是关于PSL（n，R）的Orbifold群的Hitchin分支的一般情况。我们已经证明了一些初步的结果，即Hitchin分支同胚于向量空间，并且得到了它们的维数公式，这一研究将进一步发展，以便更好地理解这些Hitchin分支。最有趣的情况是“小”轨道褶皱的情况，一些轨道褶皱的希钦分量具有非常小的维度，例如0，1或2。我们可以对0维的所有希钦分量进行分类，这非常有趣，因为它给出了令人惊讶的刚性示例。给出了三维流形上的真实的射影结构的变形空间的一些应用，我们可以找到Seifert纤维三维流形允许刚性真实的射影结构的例子，一维Hitchin分支的情况也很有趣，因为这些分支的几何很容易理解。此外，它们是研究退化现象的一个很好的候选者：当一个表示序列发散时会发生什么？此外，几何坐标的这些Hitchin组件orbifolds将开发，一个适当的推广的福克-Goncharov坐标。拥有显式坐标对于理解这些变形空间非常有用。

Teichmüller spaces parametrise the set of hyperbolic structures on a surface. They can also be seen as spaces of discrete and faithful representations of the fundamental group of the surface in PSL(2,R). These representations are called Fuchsian representations.Higher Teichmüller Theory tries to generalise the theory of Teichmüller spaces replacing the group PSL(2,R) with another Lie group G. In this project G will be usually PSL(n,R). Hitchin representations are a special kind of representations of the surface group in PSL(n,R), having similar properties to the Fuchsian representations. The sets of such representations are called Hitchin components, and they generalise Teichmüller spaces.In this project the study of Hitchin components will be generalised to a more general family of groups, namely the fundamental groups of 2-dimensional orbifolds. This is a large family of groups including for example 2-dimensional hyperbolic Coxeter groups. The case of PSL(2,R) was already studied by Thurston, who introduced hyperbolic structures on orbifolds and studied their Teichmüller spaces. The case of PSL(3,R) was studied by Choi and Goldman. This project is about the general case of Hitchin components of orbifold groups for PSL(n,R). We already proved some preliminary results, the Hitchin components are homeomorphic to a vector space, and we have a formula for their dimensions.This study will be further developed to get a better understanding of these Hitchin components. The most interesting cases are the cases of "small" orbifolds, some orbifolds whose Hitchin components have very small dimensions, say 0, 1 or 2. We can classify all the Hitchin components of dimension 0, this is very interesting because it gives surprising examples of rigidity. Some applications to the deformation spaces of real projective structures on 3-manifolds are given, we can find examples of Seifert fibered 3-manifolds admitting a rigid real projective structure.The case of Hitchin components of dimension 1 is also very interesting, because the geometry of these components can be easy to understand. Moreover, they are a very good candidate to study phenomena of degeneration: what happens when a sequence of representations diverges?Moreover, geometric coordinates on these Hitchin components for orbifolds will be developed, a suitable generalization of the Fock-Goncharov coordinates. Having explicit coordinates can be very useful to understand these deformation spaces.