Rigidity of the saddle connection complex. Issue 3 (30th June 2022) - Disarlo, Valentina; Randecker, Anja; Tang, Robert
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 ′ ) $φ \colon \mathcal {A}(S, q) → \mathcal {A}(S^{′ }, q^{′ })$ between saddle connection complexes is induced by an affine diffeomorphism F : ( S, q ) → ( S ′, q ′ ) $F \colon (S, q) → (S^{′ }, q^{′ })$ . 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.