AI 中文总结
该研究明确群的Gorenstein同调与上同调维数的关系，刻画LH𝔉类中低维群，揭示virtually可解群Hirsch长度与Gorenstein维数的关联，还探讨相关群代数模的Gorenstein维数及应用。
AI 中文摘要
我们研究群的Gorenstein同调维数与上同调维数之间的关系。证明了对任意环R上每个FP∞型模，其Gorenstein投射维数与Gorenstein平坦维数始终相等；特别地，对交换环k上每个FP∞型群G，有等式Gcd_kG = Ghd_kG。对Kropholler类LH𝔉中的非局部有限群，通过在具有有限稳定子群的树上的作用，刻画了Gorenstein（上）同调维数为1的群。证明了virtually可解群的Hirsch长度决定其Gorenstein（上）同调维数，即使群含挠元也成立，即Ghd_ℤG = h(G)；若G可数，则h(G) ≤ Gcd_ℤG ≤ h(G)+1。还得到了有限Hirsch长度的初等可和群的若干推论，最后研究了含挠元的特定群代数上模的Gorenstein维数，并讨论其在真作用分类空间中的应用。
英文摘要
We study the relation between the Gorenstein homological and the Gorenstein cohomological dimension of groups. We prove that for every module of type $FP_\infty$ over any ring the Gorenstein projective and the Gorenstein flat dimensions are always equal to each other. In particular, we have an equality $\mathrm{Gcd}_kG=\mathrm{Ghd}_kG$ for every group $G$ of type $FP_\infty$ over a commutative coefficient ring $k$. For non-locally-finite groups in Kropholler's class ${\scriptstyle\mathbf{LH}}\mathfrak F$, we characterize the groups of Gorenstein (co)homological dimension one, in terms of actions on trees with finite stabilizers. The Hirsch length of a virtually soluble group $G$ determines its Gorenstein (co)homological dimension, even when $G$ has torsion: We show that $\mathrm{Ghd}_\mathbb{Z}G =h(G)$ and, if $G$ is countable, $h(G)\leq\mathrm{Gcd}_\mathbb{Z}G\leq h(G)+1$. Some consequences for elementary amenable groups of finite Hirsch length are also obtained. Finally, we investigate the Gorenstein dimension of modules over certain algebras of groups with torsion and discuss applications to classifying spaces for proper actions.
