Relating two genus 0 problems of John Thompson

Abstract

Excluding a precise list of groups like alternating, symmetric, cyclic and dihedral, from 1st year algebra (§7.2.3), we expect there are only finitely many monodromy groups of primitive genus 0 covers. Denote this nearly proven genus 0 problem as Problem0g=02. We call the exceptional groups 0-sporadic. Example: Finitely many Chevalley groups are 0-sporadic. A proven result: Among polynoinicil 0-sporadic groups, precisely three produce covers falling in nontrivial reduced families. Each (miraculously) defines one natural genus 0 Q cover of the 7-line. The latest Nielsen class techniques apply to these dessins d'enfant to see their subtle arithmetic and interesting cusps. Jolm Thompson earlier considered another genus 0 problem: To find 6- functions uniformizing certain genus 0 (near) modular curves. We call this Problem0g=01. We pose uniformization problems for y-line covers in two cases. First: From the three 0-sporadic examples of Proble m0g=02. Second: From finite collections of genus 0 curves with aspects of Problem0g=01. © 2005 Springer Science + Business Media, Inc.