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We develop Fenchel–Nielsen coordinates for representations of surface groups into $${{\mathrm{Sp}}}(2n,{\mathbb {R}})$$ Sp ( 2 n , R ) with maximal Toledo invariant. Analogous to classical Fenchel–Nielsen coordinates on the Teichmüller space they consist of a parametrization of representations of the fundamental group of a pair of pants and a careful investigation of the gluing. As applications...
We show that there is a unique functorial way to extend the classical four-point cross ratio on the unit circle to a family of four-point invariants on Shilov boundaries of bounded symmetric domains of tube type. These generalized cross ratios can be used to estimate translation lengths of a large class of isometries of the underlying bounded symmetric domains. They can also be used to associate with...
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