Farey Maps, Spectra and Integer Continued Fractions

We examine the structure of Farey maps, a class of graph embeddings on surfaces that have received significant attention recently. When the Farey graph is embedded in the hyperbolic plane it induces a tessellation by ideal triangles. Farey maps are the quotients of this tessellation by the principal congruence subgroups of the modular group. We describe how the Farey maps of different levels are related to each other through regular coverings and parallel products, and use this to find their complete spectra. We then generalise Farey maps to include those defined by non–principal congruence subgroups of the modular group, finding their spectra and diameter. We also examine a similar class of maps defined by Hecke groups, again obtaining results for their spectra and diameter. Most of this work is the subject of [63], which has been published in Acta Mathematica Universitatis Comenianae. Fundamental to the theory of continued fractions is the fact that every infinite continued fraction with positive integer coefficients converges; however, this is not so if the coefficients are integers which are not necessarily positive. We show that integer continued fractions can be represented as paths on the Farey graph, and use this to develop a simple test that determines whether an integer continued fraction converges or diverges. In addition, for convergent continued fractions, the test specifies whether the limit is rational or irrational. This work, carried out jointly with Ian Short, is the subject of [57], which has been published in the Proceedings of the American Mathematical Society. Finally further work is described, including practical applications of our spectral results, and a search for interesting expansions of real numbers as generalised continued fractions