Ph.D Thesis

Ph.D StudentPinsky Tali
SubjectKnotted Geodesics on Hecke Triangles
DepartmentDepartment of Mathematics
Supervisor PROF. Amos Nevo
Full Thesis textFull thesis text - English Version


The general object of this thesis are closed geodesics on hyperbolic surfaces. These are the periodic orbits of the geodesic flow on the surface, which is defined on the unit tangent bundle to that surface. We analyze some of these three dimensional flows, using a tool called a Template for the flow. Templates were first studied by Birman and Williams, and are a way to encode the dynamics of the flow on a two dimensional object, which is much easier to analyze.

We first construct templates for geodesic flows on an infinite family of Hecke triangle groups. Our results generalize those of Ghys, who constructed a template for the modular flow in the complement of the trefoil knot in $S^3$.  A significant difficulty that arises in any attempt to go beyond the modular flow is the fact that for other Hecke triangles the geodesic flow cannot be viewed as a flow on a complement of a knot in $S^3$, and one is led to consider embeddings into lens spaces. Our final result is  an explicit description of a single ``Hecke template" which contains all other templates we construct, allowing a topological study of the periodic orbits of different Hecke triangle groups all at once.

We then prove that all closed primitive geodesics on such orbifolds are prime knots, in the appropriate unit tangent bundle, as well as in the closed three manifold obtained from the unit tangent bundle by the Dehn filling of the cusp of Euler number zero. In particular, we prove this for the modular knots.