论文标题
极限组的虚拟同源性和直接产品的刚性刚度
Virtual homology of limit groups and profinite rigidity of direct products
论文作者
论文摘要
我们表明，当$ g $是免费的，免费的Abelian或封闭式封闭式的基本组时，实际生成有限生成的，剩余的$ G $的虚拟第二Betti数是有限的。我们还在更高维度上证明了类似的陈述。然后，我们开发涉及Pro-P $组等级梯度的技术，使我们能够识别直接的产品分解。结合了这些想法，我们表明，自由和表面群体的直接产品在有限呈现的，剩余的群体中有限僵硬，部分解决了布里森的猜想。我们获得的其他推论包括确认梅尔尼科夫的表面组猜想，以及对封闭的尺寸的封闭的非球形歧管的描述，至少$ 5 $，具有残留的免费基本组。
We show that the virtual second Betti number of a finitely generated, residually free group $G$ is finite if and only if $G$ is either free, free abelian or the fundamental group of a closed surface. We also prove a similar statement in higher dimensions. We then develop techniques involving rank gradients of pro-$p$ groups, which allow us to recognise direct product decompositions. Combining these ideas, we show that direct products of free and surface groups are profinitely rigid among finitely presented, residually free groups, partially resolving a conjecture of Bridson's. Other corollaries that we obtain include a confirmation of Mel'nikov's surface group conjecture in the residually free case, and a description of closed aspherical manifolds of dimension at least $5$ with a residually free fundamental group.
