Automorphisms of the k-curve graph

Shuchi Agrawal; Tarik Aougab; Yassin Chandran; Marissa Loving; J. Robert Oakley; Roberta Shapiro; Yang Xiao

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arXiv · 1912.07666

Automorphisms of the k-curve graph

Shuchi Agrawal·Tarik Aougab·Yassin Chandran·Marissa Loving·J. Robert Oakley·Roberta Shapiro·Yang Xiao

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Abstract

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.

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BibTeX RIS

Shuchi Agrawal, Tarik Aougab, Yassin Chandran, Marissa Loving, J. Robert Oakley, Roberta Shapiro, Yang Xiao. 2019-12-16. Automorphisms of the k-curve graph. https://arxiv.org/abs/1912.07666

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