Almost commensurability of 3-dimensional Anosov flows

TL;DR

This paper demonstrates that suspensions of torus automorphisms and geodesic flows on hyperbolic 2-orbifolds are nearly equivalent in a topological sense, after finite modifications and coverings, revealing deep connections among 3D Anosov flows.

Contribution

It establishes the almost commensurability of broad classes of 3D Anosov flows, unifying their topological classifications under finite modifications and coverings.

Findings

All suspensions of torus automorphisms are almost commensurable.

All geodesic flows on hyperbolic 2-orbifolds are almost commensurable.

These classes are pairwise almost commensurable.

Abstract

Two flows are topologically almost commensurable if, up to removing finitely many periodic orbits and taking finite coverings, they are topologically equivalent. We prove that all suspensions of automorphisms of the 2-dimensional torus and all geodesic flows on unit tangent bundles to hyperbolic 2-orbifolds are pairwise almost commensurable.