Automorphisms of the k-Curve Graph

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 satisfying $\mathit{k}\le |\mathit{\chi }(\mathit{S})|-512$. We prove the same result for the so-called systolic complex, a variant of the curve graph with many complete subgraphs coming from interesting collections of systoles with respect to a hyperbolic metric. This resolves a conjecture of Schmutz Schaller.