Uniform Hyperbolicity of the Graphs of Curves

Let $\mathcal{C}(S_{g,p})$ denote the curve complex of the closed orientable surface of genus $g$ with $p$ punctures. Masur-Minksy and subsequently Bowditch showed that $\mathcal{C}(S_{g,p})$ is $\delta$-hyperbolic for some $\delta=\delta(g,p)$. In this paper, we show that there exists some $\delta>0$ independent of $g,p$ such that the curve graph $\mathcal{C}_{1}(S_{g,p})$ is $\delta$-hyperbolic. Furthermore, we use the main tool in the proof of this theorem to show uniform boundedness of two other quantities which a priori grow with $g$ and $p$: the curve complex distance between two vertex cycles of the same train track, and the Lipschitz constants of the map from Teichm\"{u}ller space to $\mathcal{C}(S)$ sending a Riemann surface to the curve(s) of shortest extremal length.Comment: 19 pages, 2 figures. This is a second version, revised to fix minor typos and to make the end of the main proof more understandabl

Small intersection numbers in the curve graph

Let $S_{g,p}$ denote the genus $g$ orientable surface with $p \ge 0$ punctures, and let $\omega(g,p)= 3g+p-4$. We prove the existence of infinitely long geodesic rays $\left\{v_{0},v_{1}, v_{2}, ...\right\}$ in the curve graph satisfying the following optimal intersection property: for any natural number $k$, the endpoints $v_{i},v_{i+k}$ of any length $k$ subsegment intersect $O(\omega^{k-2})$ times. By combining this with work of the first author, we answer a question of Dan Margalit.Comment: 13 pages, 6 figure