On Covering Klein’s Curve and Generating Projective Groups

An important event in the history of Riemann surface theory was F. Klein’s investigation [6] of the principal congruence subgroup at level seven of the modular group and his subsequent discovery of the famous quartic curve x3y + y3z + z3x = 0. This genus 3 curve was soon put to good use. A. Hurwitz [5] showed that a compact Riemann surface of genus g has no more than 84(g — 1) conformal homeomorphisms and cited Klein’s curve to show that this bound is attainable. It is an easy consequence of Hurwitz’s work that there is a one-to-one correspondence between conformal equivalence classes of compact Riemann surfaces that attain the upper bound and normal subgroups of finite index of $$ \left( {2,3,7} \right)\;{\rm{: = }}\left\langle {\left. {x,y:{x^2}\;{\rm{ = }}{y^3}\;{\rm{ = }}{{(xy)}^7} = \;1} \right\rangle .} \right. $$ Factors of (2,3,7) are therefore called Hurwitz groups. In [2], all such normal subgroups were obtained whose factor is an extension of an Abelian group by PSL₂(1). In other words all Abelian covers of Klein’s surface by compact Riemann surfaces exhibiting Hurwitz’s upper bound were obtained. Here a new infinite family of Hurwitz groups is given whose members act on covers of Klein’s curve. Further, matrix representations of certain groups from [2] are obtained and used to decide precisely when the extension splits.