论文标题
关于自由和表面组的刚性刚度
On the profinite rigidity of free and surface groups
论文作者
论文摘要
令$ s $是免费组或封闭双曲线表面的基本组。我们表明，如果$ g $是有限生成的剩余-P $组，则与$ s $相同的pro-p $完成，则免费的两个生成子组是免费的。这是Baumslag的类似结果（并给出了新的证明）。我们的论点取决于以下新成分：如果$ g $是一个残留的 - （无扭转的nilpotent）组，而$ h \ leq g $实际上是多环切子组，那么$ h $ nilpotent是nilpotent，pro-$ p $ p $ p $ g $ to $ g $ of $ h $ h $ h $ pro-p $ p $ p $ $ p $ topology of $ h $ po $ tocal。然后，我们研究对刚性刚性的应用。 Remeslennikov猜想，有限生成的剩余$ g $，带有profinite完成$ \ widehat g \ cong \ widehat s $一定是$ g \ cong s $。当$ g $属于一类$ \ mathcal {h} _ {ab} $的类别时，我们确认了这一点，该组具有有限的Abelian层次结构，从有限生成的无剩余组开始。这加强了威尔顿的先前结果，该结果依赖于双曲线假设。最后，我们证明了$ s \ times {\ mathbb z}^n $在有限生成的剩余组中非常刚性。
Let $S$ be either a free group or the fundamental group of a closed hyperbolic surface. We show that if $G$ is a finitely generated residually-$p$ group with the same pro-$p$ completion as $S$, then two-generated subgroups of $G$ are free. This generalises (and gives a new proof of) the analogous result of Baumslag for parafree groups. Our argument relies on the following new ingredient: if $G$ is a residually-(torsion-free nilpotent) group and $H\leq G$ is a virtually polycyclic subgroup, then $H$ is nilpotent and the pro-$p$ topology of $G$ induces on $H$ its full pro-$p$ topology. Then we study applications to profinite rigidity. Remeslennikov conjectured that a finitely generated residually finite $G$ with profinite completion $\widehat G\cong \widehat S$ is necessarily $G\cong S$. We confirm this when $G$ belongs to a class of groups $\mathcal{H}_{ab}$ that has a finite abelian hierarchy starting with finitely generated residually free groups. This strengthens a previous result of Wilton that relies on the hyperbolicity assumption. Lastly, we prove that the group $S\times {\mathbb Z}^n$ is profinitely rigid within finitely generated residually free groups.
