Templates for geodesic flows

TL;DR

This paper generalizes templates for geodesic flows from modular to Hecke triangle groups, constructing explicit homeomorphisms into lens spaces to analyze periodic orbits topologically.

Contribution

It introduces a geometric construction of homeomorphisms from unit tangent bundles of orbifolds into lens spaces, enabling template computation for geodesic flows.

Findings

Templates for geodesic flows on Hecke triangle groups are constructed.

Homeomorphisms into lens spaces are explicitly described.

Tools for topological analysis of periodic orbits are developed.

Abstract

The fact that the modular template coincides with the Lorenz template, discovered by Ghys, implies modular knots have very peculiar properties. We obtain a generalization of these results to other Hecke triangle groups. In this context, the geodesic flow can never be seen as a flow on a subset of $S^3$, and one is led to consider embeddings into lens spaces. We will geometrically construct homeomorphisms from the unit tangent bundles of the orbifolds into the lens spaces, elliminating the need for elliptic functions. Finally we will use these homeomorphisms to compute templates for the geodesic flows. This offers a tool for topologically investigating their otherwise well studied periodic orbits.