Liftable automorphisms of right-angled Artin groups

Abstract

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

29 pages, 4 figures