Let C(Sg,p) denote the curve complex of the closed orientable surface of genus g with p punctures. Masur-Minksy and subsequently Bowditch showed that C(Sg,p) is δ-hyperbolic for some δ = δ(g, p). In this paper, we show that there exists some δ > 0 independent of g, p such that the curve graph C1(Sg,p) is δ-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 Teichmüller space to C(S) sending a Riemann surface to the curve(s) of shortest extremal length.Note that C(S) is quasi-isometric to C 1 (S), however the quasi-constants depend on the underlying surface. Therefore Theorem 1.1 does not imply the uniform hyperbolicity of C(S). We also note that it has recently come to the author's attention that Brian Bowditch has independently obtained the

Let $S_{g}$ denote the closed orientable surface of genus $g$. We construct exponentially many mapping class group orbits of pairs of simple closed curves which fill $S_{g}$ and intersect minimally, by showing that such orbits are in correspondence with the solutions of a certain permutation equation in the symmetric group. Next, we demonstrate that minimally intersecting filling pairs are combinatorially optimal, in the sense that there are many simple closed curves intersecting the pair exactly once. We conclude by initiating the study of a topological Morse function $\mathcal{F}_{g}$ over the Moduli space of Riemann surfaces of genus $g$, which, given a hyperbolic metric $\sigma$, outputs the length of the shortest, minimally intersecting filling pair for the metric $\sigma$. We completely characterize the global minima of $\mathcal{F}_{g}$, and using the exponentially many mapping class group orbits of minimally intersecting filling pairs that we construct in the first portion of the paper, we show that the number of such minima grow at least exponentially in $g$.Comment: 26 pages, 11 figure

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Abstract. Let γ be an essential closed curve with at most k self-intersections on a surface S with negative Euler characteristic. In this paper, we construct a hyperbolic metric ρ for which γ has length at most M · √ k, where M is a constant depending only on the topology of S. Moreover, the injectivity radius of ρ is at least 1/(2 √ k). This yields linear upper bounds in terms of self-intersection number on the minimum degree of a cover to which γ lifts as a simple closed curve (i.e. lifts simply). We also show that if γ is a closed curve with length at most L on a cusped hyperbolic surface S, then there exists a cover of S of degree at most N ·L·e L/2 to which γ lifts simply, for N depending only on the topology of S.