New Classes of Quasigeodesic Anosov Flows in $3$-Manifolds

TL;DR

This paper introduces a new class of quasigeodesic Anosov flows in 3-manifolds, expanding the known examples beyond suspension flows and providing a novel method to establish quasigeodesicity.

Contribution

It presents the first examples of quasigeodesic Anosov flows on non-Seifert, non-solvable, non-hyperbolic 3-manifolds and develops a new technique for proving quasigeodesicity.

Findings

New class of quasigeodesic Anosov flows identified

First examples on complex 3-manifolds beyond known types

A novel method for establishing quasigeodesic behavior

Abstract

Quasigeodesic behavior of flow lines is a very useful property in the study of Anosov flows. Not every Anosov flow in dimension three is quasigeodesic. In fact up to orbit equivalence, the only previously known examples of quasigeodesic Anosov flows were suspension flows. In this article, we prove that a new class of examples are quasigeodesic. These are the first examples of quasigeodesic Anosov flows on three manifolds that are neither Seifert, nor solvable, nor hyperbolic. In general, it is very hard to show that a given flow in quasigeodesic, and in this article we provide a new method to prove that an Anosov flow is quasigeodesic.