Cutting Sequences for Geodesic Flow on the Modular Surface and Continued Fractions

Abstract

This paper describes the cutting sequences of geodesic flow on the modular surface H/PSL(2,Z) with respect to the standard fundamental domain F = {z=x+iy: -1/2 <= x <= 1/2 and |z|>=1} of PSL(2,Z). The cutting sequence for a vertical geodesic {\theta+it: t > 0} is related to a one-dimensional continued fraction expansion for \theta, called the one-dimensional Minkowski geodesic continued fraction (MGCF) expansion, which is associated to a parametrized family of reduced bases of a family of 2-dimensional lattices. The set of cutting sequences for all geodesics forms a two-sided shift in a symbol space {L,R,J} which has the same set of forbidden blocks as for vertical geodesics. We show that this shift is not a sofic shift, and that it characterizes the fundamental domain F up to an isometry of the hyperbolic plane H. We give conversion methods between the cutting sequence for the vertical geodesic {\theta+it: t > 0}, the MGCF expansion of \theta and the additive ordinary continued fraction (ACF) expansion of \theta. We show that the cutting sequence and MGCF expansions can each be computed from the other by a finite automaton, and the ACF expansion of \theta can be computed from the cutting sequence for the vertical geodesic \theta+it by a finite automaton. However, the cutting sequence for a vertical geodesic cannot be computed from the ACF expansion by any finite automaton, but there is an algorithm to compute its first l symbols when given as input the first O(l) symbols of the ACF expansion, which takes time O(l^2) and space O(l).