Rigidity of the saddle connection complex
Rigidity of the saddle connection complex
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DOI：
10.1112/topo.12242
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发表时间：
2018-10
影响因子：
1.1
通讯作者：
Valentina Disarlo;Anja Randecker;Robert L. Tang
中科院分区：
文献类型：
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作者：
Valentina Disarlo;Anja Randecker;Robert L. Tang
对于一个半平移曲面$(S,q)$，相关的鞍连接复形$\mathcal{A}(S,q)$是一个单纯复形，其中顶点是$(S,q)$上的鞍连接，单形由两两不相交的鞍连接集合张成。这个复形可自然地视为弧复形的一个诱导子复形。我们证明，鞍连接复形之间的任何单纯同构$\phi:\mathcal{A}(S,q)\to\mathcal{A}(S',q')$是由一个仿射微分同胚$F:(S,q)\to(S',q')$诱导的。特别地，这表明鞍连接复形是半平移曲面的仿射等价类的一个完全不变量。在我们的证明过程中，我们发展了几个具有独立意义的组合判别准则，用于检测半平移曲面上的各种几何对象。
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.