The saddle connection complex

A half-translation surface is given by a quadratic differential on a closed Riemann surface. It can be visualized by polygons in the plane whose edges are identified via translations and reflection-translations. The geometric properties of a half-translation surface can be captured in the saddle connection complex. The vertices of this complex are saddle connections (i.e. geodesic segments that connect a zero or pole to a zero or pole) and the simplices are formed by saddle connections that are non-intersecting. I will explain properties of the saddle connection complex, in particular the following rigidity result: Every simplicial isomorphism between the saddle connection complexes of two half-translation surfaces is induced by an affine diffeomorphism between the underlying surfaces. The talk is based on joint work with Valentina Disarlo and Robert Tang.

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