I'm wondering if anyone knows of a quick proof or reference of the following fact:
Let $S$ be a compact hyperbolic surface and $l > 0$. Then there exists a finite covering space $S' \rightarrow S$ such that every non-contractible closed curve in $S'$ has length at least $l$.
It sounds like folklore/well-known, but I can't find a specific reference. Many thanks in advance!
This is implied by the theorem that the fundamental group $\pi_1(S)$ is residually finite, which means that for any nontrivial element $\gamma \in \pi_1(S)$ there is a finite index normal subgroup of $\pi_1(S)$ which does not contain $\gamma$.
To apply this theorem, list the finitely many closed geodesics of length $<l$, and let $\gamma_1,\ldots,\gamma_n$ be elements of $\pi_1(S)$ representing each such geodesic. Net $N_i$ be a finite index normal subgroup not containing $\gamma_i$, and let $N = \cap_{i=1}^n N_i$. It follows that $N$ is a finite index normal subgroup not containing any of the $\gamma_i$'s. Let $q : S' \mapsto S$ be the covering map corresponding to $N$. Lifting the hyperbolic structure on $S$ to the unique one on $S'$ for which $q$ is locally geodesic, it follows that the systole of $S'$ is $\ge l$.