Triangle Group - Applications

Applications

Triangle groups arise in arithmetic geometry. The modular group is generated by two elements, S and T, subject to the relations S² = (ST)³ = 1 (no relation on T), is the rotational triangle group (2,3,∞) and maps onto all triangle groups (2,3,n) by adding the relation Tn = 1. More generally, the Hecke group H_q is by two elements, S and T, subject to the relations S² = (ST)q = 1 (no relation on T), is the rotational triangle group (2,q,∞), and maps onto all triangle groups (2,q,n) by adding the relation Tn = 1 the modular group is the Hecke group H₃. In Grothendieck's theory of dessins d'enfants, a Belyi function gives rise to a tessellation of a Riemann surface by reflection domains of a triangle group.

All 26 sporadic groups are quotients of triangle groups, of which 12 are Hurwitz groups (quotients of the (2,3,7) group).