On Covering Klein’s Curve and Generating Projective Groups

Abstract

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 x 3 y + y 3 z + z 3 x = 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 PSL 2(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.