Geometrical Markov coding of geodesics on surfaces of constant negative curvature

A natural geometrical representation of the geodesic flow on a surface M of constant negative curvature is given in which the base transformation is the shift on a (finite type) space of shortest words relative to a fixed generating set for π 1 ( M ) and the height function is the hyperbolic distance across a fundamental region for π 1 ( M ). This representation is obtained by comparing cutting sequences on M with generalised continued fraction expansions of endpoints on ℝ