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 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.
- Publication:
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arXiv e-prints
- Pub Date:
- December 2019
- DOI:
- 10.48550/arXiv.1912.07666
- arXiv:
- arXiv:1912.07666
- Bibcode:
- 2019arXiv191207666A
- Keywords:
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- Mathematics - Geometric Topology;
- Mathematics - Combinatorics
- E-Print:
- 33 pages, 23 figures, 1 table. Incorporated referee comments. To appear in the Michigan Mathematical Journal