TITLE: injectivity radius of hyperbolic surface
QUESTION [6 upvotes]: Given a positive real number $l$. Does there exist a closed hyperbolic surface $X$ so that injectivity radius not less than $l$?

REPLY [8 votes]: Let $Y$ be a compact hyperbolic surface.  There are only finitely many closed geodesics in $Y$ whose lengths are less that $\ell$.  Since $\pi_1(Y)$ is residually finite, there is a normal subgroup of finite index in $\pi_1(Y)$ that does not contain anything in the conjugacy classes corresponding to these geodesics. The corresponding finite covering space $X$ of $Y$ has no closed geodesics of length less than $\ell$.