Hyperbolic systole (${\mathrm{sys}}_{\mathrm{hyp}}(X)$)

The infimum of nonconstant closed geodesic lengths in curvature $-1$, with value $+\infty$ if there are none. On a closed surface with a hyperbolic metric this is a positive attained minimum, and a minimizing geodesic is simple. For fixed closed genus $g\ge 2$, a common positive lower bound gives relative compactness in the unmarked moduli space of Riemann surfaces by Mumford's compactness theorem.