交比、平移长度与极大表示
Cross ratios, translation lengths and maximal representations
摘要：
本文定义了一类管型有界对称域的Shilov边界的四点不变量，推广了单位圆上经典的四点交比.这种推广，这是基于一个类似的建设Clerc和Errsted，是函子和良好的表现下的产品，这两个属性确定我们的扩展唯一。我们的广义交比可以用来估计平移长度的一个大类的等距的基础有界对称域。我们的主要应用涉及极大表示的曲面群厄米目标。对于任何这样的表示，我们可以通过沿着适当的极限曲线拉回我们的广义交比来构造Labourie意义下的圆上的严格交比。在这种情况下，我们的翻译长度估计，然后意味着最大表示与厄米目标是良好的位移，这意味着特别是映射类群的行动上的模空间的最大表示到厄米李群是适当的。
英文摘要：
We define a family of four-point invariants for Shilov boundaries of bounded symmetric domains of tube type, which generalizes the classical four-point cross ratio on the unit circle. This generalization, which is based on a similar construction of Clerc and Ørsted, is functorial and well-behaved under products; these two properties determine our extension uniquely. Our generalized cross ratios can be used to estimate translation lengths of a large class of isometries of the underlying bounded symmetric domain. Our main application concerns maximal representations of surface groups with Hermitian target. For any such representation we can construct a strict cross ratio on the circle in the sense of Labourie via pullback of our generalized cross ratio along a suitable limit curve. In this context our translation length estimates then imply that maximal representations with Hermitian target are well-displacing; this implies in particular that the action of the mapping class group on the moduli space of maximal representations into a Hermitian Lie group is proper.
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