Automorphisms of the k-Curve Graph
Automorphisms of the k-Curve Graph
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DOI：
10.1307/mmj/20205929
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发表时间：
2019-12
影响因子：
0.9
通讯作者：
Shuchi Agrawal;Tarik Aougab;Yassin Chandran;Marissa Loving;J. Oakley;R. Shapiro;Yang Xiao
中科院分区：
文献类型：
--
作者：
Shuchi Agrawal;Tarik Aougab;Yassin Chandran;Marissa Loving;J. Oakley;R. Shapiro;Yang Xiao
给定自然数 k 和有限类型的可定向曲面 S，将 k 曲线图定义为这样的图，其顶点对应于 S 上基本简单闭合曲线的同位素类，边对应于允许最多相交 k 次的代表的此类曲线对。我们证明了曲面 S 的 k 曲线图的自同构群对于所有 k 相对于 S 的欧拉特性而言足够小，与扩展映射类群同构。我们证明了所谓的收缩复形（曲线图的一种变体）的相同结果，其完整子图编码了任何收缩集合相对于双曲度量的交集模式。这解决了施穆茨·夏勒的一个猜想。
Given a natural number k and an orientable surface S of finite type, define the k-curve graph to be the graph with vertices corresponding to isotopy classes of essential simple closed curves on S and with edges corresponding to pairs of such curves admitting representatives that intersect at most k times. We prove that the automorphism group of the k-curve graph of a surface S is isomorphic to the extended mapping class group for all k sufficiently small with respect to the Euler characteristic of S. We prove the same result for the so-called systolic complex, a variant of the curve graph whose complete subgraphs encode the intersection patterns for any collection of systoles with respect to a hyperbolic metric. This resolves a conjecture of Schmutz Schaller.