Rigidity of the saddle connection complex

Valentina Disarlo, Anja Randecker*, Robert Tang

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


For a half-translation surface (Formula presented.), the associated saddle connection complex (Formula presented.) is the simplicial complex where vertices are the saddle connections on (Formula presented.), 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 (Formula presented.) between saddle connection complexes is induced by an affine diffeomorphism (Formula presented.). 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.

Original languageEnglish
Pages (from-to)1248-1310
Number of pages63
JournalJournal of Topology
Issue number3
Publication statusPublished - Sept 2022


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