关于Bestvina-Mess公式
On Bestvina-Mess Formula
摘要：
Besthold和Mess [BM]证明了无挠的双曲群$$ \dim_L\partialΓ=cd_LΓ-1 $$的上同调维数与其边界$\partialΓ$的上同调维数之间的一个显著关系式.在[Be] Besthold介绍了一个概念的$\sZ$ -结构的离散群，并注意到他的公式适用于所有无挠的群的$\sZ$ -结构。Bestern的$\sZ$ -结构的概念可以通过用几何作用代替定义中的覆盖空间作用而推广到包含挠的群。虽然Bestvina-Mess公式对于有挠的群是无效的，但我们证明了它仍然以下面的修改形式成立：{ \it对于每个PID $L$作为系数群，群$Γ$的$\sZ$ -边界的上同调维数等于其整体上同调维数} $$ \dim_L\partialΓ=gcd_L(\partialΓ). $$利用这个公式，我们证明了边界$\dim_{L}\partialΓ$的上同调维数是一个拟上同调维数。群的等距不变量
英文摘要：
Bestvina and Mess [BM] proved a remarkable formula for torsion free hyperbolic groups $$ \dim_L\partialΓ=cd_LΓ-1 $$ connecting the cohomological dimension of a group $Γ$ with the cohomological dimension of its boundary $\partialΓ$. In [Be] Bestvina introduced a notion of $\sZ$-structure on a discrete group and noticed that his formula holds true for all torsion free groups with $\sZ$-structure. Bestvina's notion of $\sZ$-structure can be extended to groups containing torsion by replacing the covering space action in the definition by the geometric action. Though the Bestvina-Mess formula trivially is not valid for groups with torsion, we show that it still holds in the following modified form: {\it The cohomological dimension of a $\sZ$-boundary of a group $Γ$ equals its global cohomological dimension for every PID $L$ as the coefficient group} $$ \dim_L\partialΓ=gcd_L(\partialΓ). $$ Using this formula we show that the cohomological dimension of the boundary $\dim_{L}\partialΓ$ is a quasi-isometry invariant of a group.
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