WoG 2024 Talk 5.6: Layne Hall - Recognising pseudo-Anosov flows without homotopic orbits

Speaker: Layne Hall
Institution: Warwick University
Website: https://warwick.ac.uk/fac/sci/maths/p...

Title: Recognising pseudo-Anosov flows without homotopic orbits
Abstract: On a three-manifold, there are rich interactions between the geometry and topology with the dynamics of a flow on the manifold. A prototypical example is the mapping torus of a pseudo-Anosov homeomorphism, where the flow `walks upwards'. Such a flow lies in an abundant class of so-called pseudo-Anosov flows. The periodic orbits of a flow are loops, and their homotopy properties play a crucial role in understanding these flows. In particular, when a pseudo-Anosov flow has no freely homotopic orbits, we know (through the work of many) a lot about the structure of the manifold and the flow. For example, the fundamental group of the manifold must be hyperbolic. The flow is also uniquely encoded by a triangulation. A natural decision problem is then: given a finite description of a pseudo-Anosov flow, determine if there are any homotopic orbits. We will motivate this problem and briefly discuss a solution.