The action of the picard group on hyperbolic 3-space and complex continued fractions

Abstract

Continued fractions have been extensively studied in number theoretic ways. These continued fractions are expressed as compositions of M¨obius maps in the Picard group PS L(2;C) that act, by Poincar´e’s extension, as isometries on H3. We investigate the Picard group with its generators and derive the fundamental domain using a direct method. From the fundamental domain, we produce an ideal octahedron, O0, that generates the Farey tessellation of H3. We explore the properties of Farey neighbours, Farey geodesics and Farey triangles that arise from the Farey tessellation and relate these to Ford spheres. We consider the Farey addition of two rationals in R as a subdivision of an interval and hence are able to generalise this notion to a subdivision of a Farey triangle with Gaussian Farey neighbour vertices. This Farey set allows us to revisit the Farey triangle subdivision given by Schmidt [44] and interpret it as a theorem about adjacent octahedra in the Farey tessellation of H3. We consider continued fraction algorithms with Gaussian integer coe cients. We introduce an analogue of Series [45] cutting sequence across H2 in H3. We derive a continued fraction expansion based on this cutting sequence generated by a geodesic in H3 that ends at the point in C that passes through O0.