Bestvina-Brady群的自同构：IA刚性、算术共轭性与有限性

中文摘要 AI 辅助

设$H_\Gamma$是与有限连通图$\Gamma$相关的Bestvina-Brady群。对于双连通的定义图，我们证明了两个结构定理。首先，限制诱导了一个同构$\mathrm{IAut}(A_\Gamma)\cong \mathrm{IAut}(H_\Gamma)$，并与Andreadakis-Johnson过滤兼容。其次，二次和三次下正则空间，加上由Bieri-Neumann-Strebel不变量检测到的分离排列，确定了一个有理关联代数$\mathscr{C}_\Gamma$。这个代数中的每一个整数秩为一的平方零元都可以由$H_\Gamma$的自同构实现，且由这些根生成的子群在$\mathrm{Aut}(H_\Gamma)$的上同调像和在$\mathscr{C}_\Gamma$的整数阶的单位群中都有有限指数。对于任意连通图，图块分解给出了$H_\Gamma$的Grushko分解。相对自由积自同构理论然后表明，$\mathrm{Aut}(H_\Gamma)$和$\mathrm{Out}(H_\Gamma)$都是有限生成的，并且相对于虚拟多项式群满足Tits替代。我们证明$\mathrm{Aut}(H_\Gamma)$是有限呈现当且仅当$\mathrm{Out}(H_\Gamma)$是有限呈现。这种等价性在没有额外假设的情况下对于更高有限性属性会失效：对于$\Gamma_m=C_m\vee K_3$且$m\geq 5$，$\mathrm{Out}(H_{\Gamma_m})$是类型$F_\infty$，而$\mathrm{Aut}(H_{\Gamma_m})$是类型$F_3$但不是$F_4$。我们还构造了一个类型$F_\infty$的Bestvina-Brady群，其自同构和外自同构群是有限生成的但不是有限呈现的，并且显示$H_{C_n}$对于$n\geq 5$不是有限呈现的，而$\mathrm{Out}(H_{C_n})$是虚拟无限循环的。

英文摘要

Let $H_Γ$ 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 $\mathrm{IAut}(A_Γ)\cong \mathrm{IAut}(H_Γ)$ 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 $\mathscr{C}_Γ$. Every integral rank-one square-zero element of this algebra is realized by an automorphism of $H_Γ$, and the subgroup generated by these roots has finite index both in the cohomological image of $\mathrm{Aut}(H_Γ)$ and in the unit group of an integral order in $\mathscr{C}_Γ$. For an arbitrary connected graph, the graph-block decomposition gives the Grushko decomposition of $H_Γ$. Relative free-product automorphism theory then implies that $\mathrm{Aut}(H_Γ)$ and $\mathrm{Out}(H_Γ)$ are finitely generated and satisfy the Tits alternative relative to virtually polycyclic groups. We prove that $\mathrm{Aut}(H_Γ)$ is finitely presented if and only if $\mathrm{Out}(H_Γ)$ is finitely presented. This equivalence fails for higher finiteness properties without additional hypotheses: for $Γ_m=C_m\vee K_3$ with $m\geq 5$, $\mathrm{Out}(H_{Γ_m})$ is of type $F_\infty$, whereas $\mathrm{Aut}(H_{Γ_m})$ 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.

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