Mumford's compactness theorem
In mathematics, Mumford's compactness theorem states that the space of compact Riemann surfaces of fixed genus g > 1 with no closed geodesics of length less than some fixed ε > 0 in the Poincaré metric is compact. It was proved by  as a consequence of a  theorem about the compactness of sets of discrete subgroups of semisimple Lie groups generalizing Mahler's compactness theorem. References

Riemann surfaces
Kleinian groups
Compactness theorems