X0 with |Aut(X0)| maximal. In this range, there only exists a Hurwitz curve in genus g=3 and g=7. Generalizations The concept of a Hurwitz surface can be generalized in several ways to a definition that has examples in all but a few genera. Perhaps the most natural is a "maximally symmetric" surface: One that cannot be continuously modified through equally symmetric surfaces to a surface whose symmetry properly contains that of the original suface. This is possible for all orientable compact genera (see above section "Automorphism groups in low genus"). See also (2,3,7) triangle group Notes References Theorems in algebraic geometry Riemann surfaces Theorems in group theory Theorems in complex geometry