Given a positive real number $l$. Does there exist a closed hyperbolic surface $X$ so that injectivity radius not less than $l$?



injectivity radius of hyperbolic surface
Buser in 1992 gives a lower bound on the Bers constant for surfaces of genus $g$, and it goes to infinity as $g$ goes to infinity. So this means there's compact hyperbolic surfaces whose injectivity radius is arbitrarily large, but you have to go to high genus to realize them. 


P. Buser. *Geometry and spectra of compact Riemann surfaces* Prog. Math. Vol 106. (1992)


The genus needs to grow like the square of the injectivity radius. Probably a good way to find this surface would be to use Fenchel-Nielsen coordinates, where the pants correspond to points on a planar square lattice and you want to connect the vertices in some (likely non-planar) tri-valent graph configuration, so that it represents pants. Then you want to connect the edges in a way so that there's no short closed loops in the resulting graph. So I'm describing a surface that's less the gluing-together of pants, but more the gluing together of quadratically-many "short shorts". 


