标题： 超极大表示从带有孔的曲面的基本群到$\mathrm{PSL}(2,\mathbb{R})$ 显示英文标题

标题： Super-maximal representations from fundamental groups of punctured surfaces to $\mathrm{PSL}(2,\mathbb{R})$

摘要： 我们研究从带有洞的球面的基本群 $\Sigma_{0,n}$ 到群 $\text{PSL} (2,\mathbb R)$ (以及它们的模空间) 的一类特殊表示，我们称之为 \emph{超极大}。 超极大表示被证明是 \emph{完全非双曲的}，也就是说，每条简单闭曲线都被映射为非双曲元素。 它们也被证明是 \emph{可几何化的} (除了可约的超极大表示)，以非常强的意义来说：对于汤姆森空间 $\mathcal T_{0,n}$ 中的任何元素，存在唯一的取值于下半平面 $\mathbb H^-$ 的全纯等变映射。 在相对特征簇中，超极大表示的组成部分被证明是紧致的，并且与复射影空间维度$n-3$在某种倍数的Fubini-Study形式下辛同构，我们显式地计算了这个倍数（这推广了Benedetto--Goldman对于四点球面的结果）。 这些是相对特征簇中唯一的紧致组成部分，$\text{PSL}(2,\mathbb R)$。 这一事实将在一篇附录论文中证明。 显示英文摘要

摘要： We study a particular class of representations from the fundamental groups of punctured spheres $\Sigma_{0,n}$ to the group $\text{PSL} (2,\mathbb R)$ (and their moduli spaces), that we call \emph{super-maximal}. Super-maximal representations are shown to be \emph{totally non hyperbolic}, in the sense that every simple closed curve is mapped to a non hyperbolic element. They are also shown to be \emph{geometrizable} (appart from the reducible super-maximal ones) in the following very strong sense : for any element of the Teichm\"uller space $\mathcal T_{0,n}$, there is a unique holomorphic equivariant map with values in the lower half-plane $\mathbb H^-$. In the relative character variety, the components of super-maximal representations are shown to be compact, and symplectomorphic (with respect to the Atiyah-Bott-Goldman symplectic structure) to the complex projective space of dimension $n-3$ equipped with a certain multiple of the Fubiny-Study form that we compute explicitly (this generalizes a result of Benedetto--Goldman for the sphere minus four points). Those are the unique compact components in relative character varieties of $\text{PSL}(2,\mathbb R)$. This latter fact will be proved in a companion paper.