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Local geometry of the $ k$-curve graph


Author: Tarik Aougab
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 32G15, 57M07, 57M50
DOI: https://doi.org/10.1090/tran/7098
Published electronically: December 29, 2017
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Abstract: Let $ S$ be an orientable surface with negative Euler characteristic. For $ k \in \mathbb{N}$, let $ \mathcal {C}_{k}(S)$ denote the k-curve graph, whose vertices are isotopy classes of essential simple closed curves on $ S$ and whose edges correspond to pairs of curves that can be realized to intersect at most $ k$ times. The theme of this paper is that the geometry of Teichmüller space and of the mapping class group captures local combinatorial properties of $ \mathcal {C}_{k}(S)$, for large $ k$. Using techniques for measuring distance in Teichmüller space, we obtain upper bounds on the following three quantities for large $ k$: the clique number of $ \mathcal {C}_{k}(S)$ (exponential in $ k$, which improves on previous bounds of Juvan, Malnič, and Mobar and Przytycki); the maximum size of the intersection, whenever it is finite, of a pair of links in $ \mathcal {C}_{k}$ (quasi-polynomial in $ k$); and the diameter in $ \mathcal {C}_{0}(S)$ of a large clique of $ \mathcal {C}_{k}(S)$ (uniformly bounded). As an application, we obtain quasi-polynomial upper bounds, depending only on the topology of $ S$, on the number of short simple closed geodesics on any unit-square tiled surface homeomorphic to $ S$.


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Additional Information

Tarik Aougab
Affiliation: Department of Mathematics, Brown University, 151 Thayer Street, Providence, Rhode Island 02902

DOI: https://doi.org/10.1090/tran/7098
Keywords: Curves on surfaces, curve systems, mapping class group, Teichm{\"u}ller space
Received by editor(s): November 16, 2015
Received by editor(s) in revised form: July 12, 2016, and October 3, 2016
Published electronically: December 29, 2017
Article copyright: © Copyright 2017 American Mathematical Society

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