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

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. In addition, we determine all $\ell^2$-Betti numbers of the outer automorphism group in the case where the defining graph is disconnected and the associated RAAG is not isomorphic to a free group. 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.