2026/07/25 by Jialin Lei
Mathematics · #math.GR
Let HΓ be the Bestvina-Brady group associated to a finite connected graph Γ. For a biconnected defining graph, we prove two structure theorems. First, restriction induces an isomorphism IAut(AΓ)≅ IAut(HΓ) 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 \mathscrCΓ. Every integral rank-one square-zero element of this algebra is realized by an automorphism of HΓ, and the subgroup generated by these roots has finite index both in the cohomological image of Aut(HΓ) and in the unit group of an integral order in \mathscrCΓ. For an arbitrary connected graph, the graph-block decomposition gives the Grushko decomposition of HΓ. Relative free-product automorphism theory then implies that Aut(HΓ) and Out(HΓ) are finitely generated and satisfy the Tits alternative relative to virtually polycyclic groups. We prove that Aut(HΓ) is finitely presented if and only if Out(HΓ) is finitely presented. This equivalence fails for higher finiteness properties without additional hypotheses: for Γm=Cm\vee K3 with m≥ 5, Out(HΓm) is of type F_∞, whereas Aut(HΓm) is of type F3 but not F4. We also construct a type-F_∞ Bestvina-Brady group whose automorphism and outer automorphism groups are finitely generated but not finitely presented, and show that HCn is not finitely presented for n≥ 5, whereas Out(HCn) is virtually infinite cyclic.