Hurwitz surface

Automorphisms of complex algebraic curves are orientation-preserving automorphisms of the underlying real surface; if one allows orientation-reversing isometries, this yields a group twice as large, of order 168(g − 1), which is sometimes of interest.

Only finitely many Hurwitz surfaces occur with each genus.

The Hurwitz surface of least genus is the Klein quartic of genus 3, with automorphism group the projective special linear group PSL(2,7), of order 84(3 − 1) = 168 = 23·3·7, which is a simple group; (or order 336 if one allows orientation-reversing isometries).

An interesting phenomenon occurs in the next possible genus, namely 14.

Here there is a triple of distinct Riemann surfaces with the identical automorphism group (of order 84(14 − 1) = 1092 = 22·3·7·13).

The sequence of allowable values for the genus of a Hurwitz surface begins