Rigidity of the saddle connection complex

Abstract

For a half-translation surface ( S , q ) $(S,q)$ , the associated saddle connection complex A ( S , q ) $\mathcal {A}(S,q)$ is the simplicial complex where vertices are the saddle connections on ( S , q ) $(S,q)$ , with simplices spanned by sets of pairwise disjoint saddle connections. This complex can be naturally regarded as an induced subcomplex of the arc complex. We prove that any simplicial isomorphism ϕ : A ( S , q ) → A ( S ′ , q ′ ) $\phi \colon \mathcal {A}(S,q) \rightarrow \mathcal {A}(S^{\prime },q^{\prime })$ between saddle connection complexes is induced by an affine diffeomorphism F : ( S , q ) → ( S ′ , q ′ ) $F \colon (S,q) \rightarrow (S^{\prime },q^{\prime })$ . In particular, this shows that the saddle connection complex is a complete invariant of affine equivalence classes of half-translation surfaces. Throughout our proof, we develop several combinatorial criteria of independent interest for detecting various geometric objects on a half-translation surface.