Local geometry of the $k$-curve graph

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 $S$.