Title : Simple geodesic on hyperbolic punctured torus
(a joint work with Chaim Goodman--Strauss)
Speaker : Yo'av Rieck (University of Arkansas)
Abstract :

Let $T$ be a complete finite area complete hyperbolic punctured torus. A geodesics on $T$ is called simple if it has no transverse self-intersection (therefore it is either an embedded copy of $\mathbb R$ or an embedded circle). McShane studied simple geodesics that exit a given cusp of $T$ and showed that they naturally correspond to a cantor set union some isolated points. He shows further that every open set complementary to the Cantor set contains exactly one isolated point and a simple geodesic corresponds to such point if and only if it returns to the cusp. In this talk, after explaining the details of this theorem we will give a very simple proof of it, that in addition shows that the set of simple cuspidal geodesics is independent of the metric: given two hyperbolic tori $T_1$ and $T_2$ there is a homeomorphism between a neighborhood of the cusp of $T_1$ and a neighborhood of the cusp of $T_2$ that takes segment of simple geodesics on $T_1$ to segments of simple geodesics on $T_2$. Time permitting we will say a word about generalizing this work to all complete hyperbolic surfaces.

pdf-file