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

Abstract

Let $H_\Gamma$ be the Bestvina-Brady group associated to a finite connected graph $\Gamma$. For a biconnected defining graph, we prove two structure theorems. First, restriction induces an isomorphism $\mathrm{IAut}(A_\Gamma)\cong \mathrm{IAut}(H_\Gamma)$ 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 $\mathscr{C}_\Gamma$. Every integral rank-one square-zero element of this algebra is realized by an automorphism of $H_\Gamma$, and the subgroup generated by these roots has finite index both in the cohomological image of $\mathrm{Aut}(H_\Gamma)$ and in the unit group of an integral order in $\mathscr{C}_\Gamma$. For an arbitrary connected graph, the graph-block decomposition gives the Grushko decomposition of $H_\Gamma$. Relative free-product automorphism theory then implies that $\mathrm{Aut}(H_\Gamma)$ and $\mathrm{Out}(H_\Gamma)$ are finitely generated and satisfy the Tits alternative relative to virtually polycyclic groups. We prove that $\mathrm{Aut}(H_\Gamma)$ is finitely presented if and only if $\mathrm{Out}(H_\Gamma)$ is finitely presented. This equivalence fails for higher finiteness properties without additional hypotheses: for $\Gamma_m=C_m\vee K_3$ with $m\geq 5$, $\mathrm{Out}(H_{\Gamma_m})$ is of type $F_\infty$, whereas $\mathrm{Aut}(H_{\Gamma_m})$ is of type $F_3$ but not $F_4$. We also construct a type-$F_\infty$ 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 $n\geq 5$, whereas $\mathrm{Out}(H_{C_n})$ is virtually infinite cyclic.