When actions of amenable groups can be lifted to the universal cover

TL;DR

This paper investigates when actions of certain amenable groups on manifolds can be lifted to their universal covers, providing conditions for liftability and analyzing specific cases like the annulus with $bZ^2$ actions.

Contribution

It establishes conditions under which group actions homotopic to the identity lift to the universal cover, and characterizes the dynamical structure of $bZ^2$ actions on the annulus.

Findings

Actions of amenable groups with torsion-free abelianization lift to the universal cover under certain conditions.

Most 2-manifolds and some 3-manifolds satisfy these liftability conditions.

Non-liftable actions on the annulus are classified up to a specific dynamical form.

Abstract

In the first part of this paper, we let $G$ be a finitely-generated amenable group such that $G/[G, G]$ is torsion-free. We suppose that $G$ acts by homeomorphisms homotopic to the identity on a manifold $M$, and give conditions on $M$ which imply that such an action must lift to an action on the universal cover $\tilde{M}$. The circle, all 2-manifolds except the open annulus, and most compact 3-manifolds satisfy these conditions. The proof uses a dynamical tool called homological rotation vectors, and Thurston's Geometrization Theorem in the latter case. On manifolds not satisfying our conditions, such actions really may fail to lift. In the second part, we try to understand the dynamical possibilities in the simplest case: $G = \mathbb{Z}^2$, and $M = \mathbb{A}$ is the open annulus. We show that if a $\mathbb{Z}^2$ action homotopic to the identity on $\mathbb{A}$ fails to lift to a $\mathbb{Z}^2$ action on the plane, and if the action satisfies one additional condition (which may not be necessary), the action is essentially similar to the one generated by $\bar{f_0}(\theta, y) = (\theta + y, y)$ and $\bar{g_0}(\theta, y) = (\theta, y + 1)$.