Building hyperbolic metrics suited to closed curves and applications to lifting simply

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