Automorphisms of the k-curve graph

TL;DR

This paper proves that the automorphism group of the k-curve graph and systolic complex of a surface is isomorphic to the extended mapping class group for small k, confirming a conjecture of Schmutz Schaller.

Contribution

It establishes the isomorphism between automorphism groups and the extended mapping class group for the k-curve graph and systolic complex, resolving a conjecture.

Findings

Automorphism group of the k-curve graph is isomorphic to the extended mapping class group for small k.

The same isomorphism holds for the systolic complex, a variant of the curve graph.

Resolves Schmutz Schaller's conjecture.

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.