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 ) 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 A(S, q) → A(S^′ , q^′ ) between saddle connection complexes is induced by an affine diffeomorphism F : ( S, q ) → ( S ′, q ′ ) F (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.