Hecke groups and continued fractions

Abstract

The Hecke groupsare Fuchsian groups of the first kind. In an interesting analogy to the use of ordinary continued fractions to study the geodesics of the modular surface, the λ-continued fractions (λF) introduced by the first author can be used to study those on the surfaces determined by the Gq. In this paper we focus on periodic continued fractions, corresponding to closed geodesics, and prove that the period of the λF for periodic has nearly the form of the classical case. From this, we give: (1) a necessary and sufficient condition for to be periodic; (2) examples of elements of ℚ(λq) which also have such periodic expansions; (3) a discussion of solutions to Pell's equation in quadratic extensions of the ℚ(λq); and (4) Legendre's constant of diophantine approximation for the Gq, that is, γq such that < γq/Q2 implies that P/Q of “reduced finite λF form” is a convergent of real α ∉ Gq(∞).