Liftable automorphisms of right-angled Artin groups

TL;DR

This paper studies the subgroup of automorphisms of right-angled Artin groups that lift through graph coverings, showing they are generated by elementary automorphisms and analyzing the structure of related lift groups.

Contribution

It characterizes liftable automorphisms of right-angled Artin groups via finite generators and describes the kernel structure of the associated lift homomorphism.

Findings

LAut(φ) is generated by Laurence's elementary automorphisms.

The kernel of the lift homomorphism is virtually a subgroup of IA(A_Λ).

Establishes a short exact sequence analogous to Birman--Hilden theory.

Abstract

Given a regular covering map $\varphi:\Lambda \to \Gamma$ of graphs, we investigate the subgroup $\operatorname{LAut}(\varphi)$ of the automorphism group $\operatorname{Aut}(A_\Gamma)$ of the right-angled Artin group $A_\Gamma$. This subgroup comprises all automorphisms that can be lifted to automorphisms of $A_\Lambda$. We first show that $\operatorname{LAut}(\varphi)$ is generated by a finite subset of Laurence's elementary automorphisms. For the subgroup $\operatorname{FAut}(\varphi)$ of $\operatorname{Aut}(A_\Lambda)$, which consists of lifts of automorphisms in $\operatorname{LAut}(\varphi)$, there exists a natural homomorphism $\operatorname{FAut}(\varphi)\to\operatorname{LAut}(\varphi)$ induced by $\varphi$. We then show that the kernel of this homomorphism is virtually a subgroup of the Torelli subgroup $\operatorname{IA}(A_\Lambda)$ and deduce a short exact sequence reminiscent of results from the Birman--Hilden theory for surfaces.