LOCAL GEOMETRY OF THE k-CURVE GRAPH

• Published in: Transactions of the American Mathematical Society 2018-04, Vol.370 (4), p.2657-2678
• Main Author: AOUGAB, TARIK
• Format: Article
• Language: English
• ISSN: 0002-9947; 1088-6850
• DOI: 10.1090/tran/7098

Let S be an orientable surface with negative Euler characteristic. For k ∈ N, let Ck(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 Ck(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 Ck(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 Ck (quasi-polynomial in k); and the diameter in C0(S) of a large clique of Ck(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.