Local geometry of the k-curve graph

Abstract

Let $S$ be an orientable surface with negative Euler characteristic. For $k\in \mathbb{N}$, let ${\mathcal{C}}_k(S)$ denote the $\text{k-curve graph}$, 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 ${\mathcal{C}}_k(S)$. 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 all previously known bounds and which is essentially sharp); 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 in ${\mathcal{C}}_k(S)$ (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 square-tiled surface homeomorphic to $S$.

20 pages; 1 figure; revised to correct typos and simplify arguments (twice)

References in corpus (1)

• Geometry of the Complex of Curves I: Hyperbolicity