Conjugate Words and Intersections of Geodesics in $H^2$

Abstract

We investigate intersections of geodesic lines in $H^2$ and in an associated tree T, proving the following result. Let M be a punctured hyperbolic torus and let $γ$ be a closed geodesic in M. Any edge of any triangle formed by distinct geodesic lines in the preimage of $γ$ in $H^2$ is shorter then $γ$. However, a similar result does not hold in the tree T. Let W be a reduced and cyclically reduced word in $π_1(M) = $. We construct several examples of triangles in T formed by distinct axes in T stabilized by conjugates of W such that an edge in those triangles is longer than L(W). We also prove that if W overlaps two of its conjugates in such a way that the overlaps cover all of W and the overlaps do not intersect, then there exists a decomposition $W = BC^kI; k > 0$, with B a terminal subword of C and I an initial subword of C.