A CLASSIFICATION OF SURDS OF NON-ARITHMETICAL GROUPS

Abstract

Definitions and classifications of reduced surds for non-arithmetical groups on the hyperbolic plane is proposed, as well as those of some congruence subgroups. The considered groups are the non-arithmetical desymmetrized group, its Γ0 congruence subgroup, and a generalized Hecke group according the choice of the commutator subgroup of the desymmetrized group. The definitions of the reduced surds are studied according to the ordering of the generators of the conjugacy subclasses of the considered groups. The construction of tori is defined as obtained after the tessellation of the Upper Hyperbolic Half Plane and discussed. Cutting trajectories and cutting-trajectories sequences are uniquely defined after the new definition of reduced surds.