Rigidity of the saddle connection complex

TL;DR

This paper proves that the saddle connection complex uniquely determines the affine equivalence class of a half-translation surface, establishing a rigidity result that links combinatorial and geometric structures.

Contribution

It demonstrates that any simplicial isomorphism between saddle connection complexes arises from an affine diffeomorphism, showing the complex's role as a complete invariant.

Findings

Saddle connection complexes are complete invariants for affine classes.

Any simplicial isomorphism between these complexes is geometric.

Develops combinatorial criteria for geometric object detection.

Abstract

For a half-translation surface (S,q), the associated saddle connection complex A(S,q) is the simplicial complex where vertices are the saddle connections on (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 $\phi : A(S,q) \to A(S',q')$ between saddle connection complexes is induced by an affine diffeomorphism $F : (S,q) \to (S',q')$. 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.