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Lifting curves simply

We provide linear lower bounds for $f_ρ(L)$, the smallest integer so that every curve on a fixed hyperbolic surface $(S,ρ)$ of length at most $L$ lifts to a simple curve on a cover of degree at most $f_ρ(L)$. This bound is independent of hyperbolic structure $ρ$, and improves on a recent bound of Gupta-Kapovich. When $(S,ρ)$ is without punctures, using work of Patel we conclude asymptotically linear growth of $f_ρ$. When $(S,ρ)$ has a puncture, we obtain exponential lower bounds for $f_ρ$.