On the ${\ell }^2$-Betti numbers and algebraic fibring of the (outer) automorphism group of right-angled Artin groups

Abstract

We compute the first ${\ell }^2$-Betti number of the automorphism and outer automorphism groups of arbitrary right-angled Artin groups (RAAGs), providing a complete characterization of when it is non-zero. We also analyse the algebraic fibring of the pure symmetric automorphism groups $\mathrm{PSA}(A_{\Gamma })$ and $\mathrm{PSO}(A_{\Gamma })$ and the virtual algebraic fibring of $\mathrm{Out}(A_{\Gamma })$ in the case when $A_{\Gamma }$ admits no non-inner partial conjugation. In the transvection-free case, we show that ${\beta }_1^{(2)}(\mathrm{Out}(A_{\Gamma }))=0$ if and only if $\mathrm{Out}(A_{\Gamma })$ virtually fibres.