Cross ratios, translation lengths and maximal representations

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 {\O}rsted, 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.