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

We compute the first ${\mathrm{\ell }}^2$-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 $\mathrm{P}\mathrm{S}\mathrm{A}(A_{\mathrm{\Gamma }})$ and $\mathrm{P}\mathrm{S}\mathrm{O}(A_{\mathrm{\Gamma }})$ and the virtual algebraic fibring of $\mathrm{O}\mathrm{u}\mathrm{t}(A_{\mathrm{\Gamma }})$ in the case when $A_{\mathrm{\Gamma }}$ admits no non-inner partial conjugation. In the transvection-free case, we show that ${\beta }_1^{(2)}(\mathrm{O}\mathrm{u}\mathrm{t}(A_{\mathrm{\Gamma }}))=0$ if and only if $\mathrm{O}\mathrm{u}\mathrm{t}(A_{\mathrm{\Gamma }})$ virtually fibres.