For a half-translation surface, we define the saddle connection complex where vertices are saddle connections and simplices are spanned by pairwise disjoint saddle connections. In a joint work with Valentina Disarlo and Robert Tang, we show that every simplicial isomorphism between the saddle connection complexes of two half-translation surfaces is induced by an affine diffeomorphism between the underlying surfaces.

In my talk, I will compare the arc complex and the saddle connection complex and I will outline our proof of the rigidity result described above.