Conjugate words and intersections of geodesics in ℍ2

Abstract

We investigate intersections of geodesic lines in [Formula: see text] and in an associated tree [Formula: see text], proving the following result. Let [Formula: see text] be a punctured hyperbolic torus and let [Formula: see text] be a closed geodesic in [Formula: see text]. Any edge of any triangle formed by distinct geodesic lines in the preimage of [Formula: see text] in [Formula: see text] is shorter than [Formula: see text]. However, a similar result does not hold in the tree T. Let [Formula: see text] be a reduced and cyclically reduced word in [Formula: see text]. We construct several examples of triangles in [Formula: see text] formed by distinct axes in [Formula: see text] stabilized by conjugates of [Formula: see text] such that an edge in those triangles is longer than [Formula: see text]. We also prove that if [Formula: see text] overlaps two of its conjugates in such a way that the overlaps cover all of [Formula: see text] and the overlaps do not intersect, then there exists a decomposition [Formula: see text], with [Formula: see text] a terminal subword of [Formula: see text] and [Formula: see text] an initial subword of [Formula: see text].