Rigidity of the saddle connection complex
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 reg...
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| Hauptverfasser: | , , |
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| Dokumenttyp: | Article (Journal) |
| Sprache: | Englisch |
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30 June 2022
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| In: |
Journal of topology
Year: 2022, Jahrgang: 15, Heft: 3, Pages: 1248-1310 |
| ISSN: | 1753-8424 |
| DOI: | 10.1112/topo.12242 |
| Online-Zugang: | Verlag, lizenzpflichtig, Volltext: https://doi.org/10.1112/topo.12242 Verlag, lizenzpflichtig, Volltext: https://onlinelibrary.wiley.com/doi/abs/10.1112/topo.12242 
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| Verfasserangaben: | Valentina Disarlo, Anja Randecker, Robert Tang |
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| 520 | |a 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. | ||
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