Automorphisms of Bestvina-Brady Groups: IA Rigidity, Arithmetic Commensurability, and Finiteness

Let $H_{\mathrm{\Gamma }}$ be the Bestvina-Brady group associated to a finite connected graph $\mathrm{\Gamma }$. For a biconnected defining graph, we prove two structure theorems. First, restriction induces an isomorphism $\mathrm{I}\mathrm{A}\mathrm{u}\mathrm{t}(A_{\mathrm{\Gamma }})\cong \mathrm{I}\mathrm{A}\mathrm{u}\mathrm{t}(H_{\mathrm{\Gamma }})$ compatible with the Andreadakis-Johnson filtrations. Second, the quadratic and cubic lower-central relation spaces, together with the separator arrangement detected by the Bieri-Neumann-Strebel invariant, determine a rational associative algebra ${\mathcal{C}}_{\mathrm{\Gamma }}$. Every integral rank-one square-zero element of this algebra is realized by an automorphism of $H_{\mathrm{\Gamma }}$, and the subgroup generated by these roots has finite index both in the cohomological image of $\mathrm{A}\mathrm{u}\mathrm{t}(H_{\mathrm{\Gamma }})$ and in the unit group of an integral order in ${\mathcal{C}}_{\mathrm{\Gamma }}$. For an arbitrary connected graph, the graph-block decomposition gives the Grushko decomposition of $H_{\mathrm{\Gamma }}$. Relative free-product automorphism theory then implies that $\mathrm{A}\mathrm{u}\mathrm{t}(H_{\mathrm{\Gamma }})$ and $\mathrm{O}\mathrm{u}\mathrm{t}(H_{\mathrm{\Gamma }})$ are finitely generated and satisfy the Tits alternative relative to virtually polycyclic groups. We prove that $\mathrm{A}\mathrm{u}\mathrm{t}(H_{\mathrm{\Gamma }})$ is finitely presented if and only if $\mathrm{O}\mathrm{u}\mathrm{t}(H_{\mathrm{\Gamma }})$ is finitely presented. This equivalence fails for higher finiteness properties without additional hypotheses: for ${\mathrm{\Gamma }}_m=C_m\vee K_3$ with $m\ge 5$, $\mathrm{O}\mathrm{u}\mathrm{t}(H_{{\mathrm{\Gamma }}_m})$ is of type $F_{\mathrm{\infty }}$, whereas $\mathrm{A}\mathrm{u}\mathrm{t}(H_{{\mathrm{\Gamma }}_m})$ is of type $F_3$ but not $F_4$. We also construct a type-$F_{\mathrm{\infty }}$ Bestvina-Brady group whose automorphism and outer automorphism groups are finitely generated but not finitely presented, and show that $H_{C_n}$ is not finitely presented for $n\ge 5$, whereas $\mathrm{O}\mathrm{u}\mathrm{t}(H_{C_n})$ is virtually infinite cyclic.