Continued fractions for a class of triangle groups

TL;DR

This paper develops continued fraction algorithms for specific Fuchsian triangle groups, linking group theory, ergodic theory, and Diophantine approximation to explore their properties and transcendence implications.

Contribution

It provides explicit forms of these triangle groups within the Hilbert modular group and introduces an interval map with ergodic properties for the continued fractions.

Findings

Explicit group forms within Hilbert modular groups

An ergodic invariant measure for the interval map

Applications to Diophantine approximation and transcendence

Abstract

We give continued fraction algorithms for a particular class of Fuchsian triangle groups. In particular, we give an explicit form of each such group that is a subgroup of the Hilbert modular group of its trace field and provide an interval map that is piecewise linear fractional, given in terms of group elements. Using natural extensions, we find an ergodic invariant measure for the interval map. We also study diophantine properties of approximation in terms of the continued fractions; and furthermore show that these continued fractions are appropriate to obtain transcendence results.