Actions of right-angled Artin groups in low dimensions

TL;DR

This survey explores how right-angled Artin groups act on low-dimensional manifolds, highlighting their subgroup structures and the constraints on smoothness of these actions across different dimensions.

Contribution

It provides a comprehensive overview of the actions of right-angled Artin groups on low-dimensional manifolds, emphasizing the interplay between algebraic properties and smoothness constraints.

Findings

Right-angled Artin groups act faithfully by $C^1$ diffeomorphisms on compact 1-manifolds.

Actions by $C^2$ diffeomorphisms on 1-manifolds are highly restricted.

In dimensions two and higher, all right-angled Artin groups act faithfully by $C^{inite}$ diffeomorphisms.

Abstract

We survey the role of right-angled Artin groups in the theory of diffeomorphism groups of low dimensional manifolds. We first describe some of the subgroup structure of right-angled Artin groups. We then discuss the interplay between algebraic structure, compactness, and regularity for group actions on one--dimensional manifolds. For compact one--manifolds, every right-angled Artin group acts faithfully by $C^1$ diffeomorphisms, but the right-angled Artin groups which act faithfully by $C^2$ diffeomorphisms are very restricted. For the real line, every right-angled Artin group acts faithfully by $C^{\infty}$ diffeomorphisms, though analytic actions are again more limited. In dimensions two and higher, every right-angled Artin group acts faithfully on every manifold by $C^{\infty}$ diffeomorphisms. We give applications of this discussion to mapping class groups of surfaces and related groups.