论文标题
有限呈现的群体的拓扑等效关系
A topological equivalence relation for finitely presented groups
论文作者
论文摘要
在本文中，我们考虑了在有限呈现的离散群体中的等效关系，该类别是在其渐近拓扑的，而不是其渐近几何形状。更确切地说，我们说，如果存在（等效地等效地）有限$ 2 $ 2 $二维的CW-Complexes $ x $ x $ x $ y $，$π_1（x）（x）\ g $ and $π_1（y $π_1（y）comcopts comcopts，则两个有限$ 2 $ 2 $ 2 $ 2 $ d， $ \ widetilde {y} $是适当的$ 2 $ - 等价。因此，这种关系比准等级的关系更粗糙。我们指出，有限呈现的$ 1 $ $ 1 $且可在无穷大的群体被分类为$ 2 $ - 等价性，其基本亲组，我们研究了这种关系与组合群体理论中某些主要结构的行为。对于$ f_n，n \ geq 3 $的组，也可以考虑（较细的）类似等价关系，该组捕获了该组的更多大规模拓扑。最后，我们特别注意这些组$ g $的类别，这些类别$ 2 $ 2 $二维的cw-complex $ x $带有$π_1（x）（x）\ g $，并且其通用封面$ \ widetilde {x} $具有适当的同型同质型类型的$ 3 $ - manifold。我们表明，如果这样的组$ g $是$ 1 $ - 可以在无穷大的情况下进行半固定，那么它是$ 2 $ - 等于$ {\ Mathbb z} \ times {\ Mathbb z} \ times {\ Mathbb z} \ times {\ mathbb z} $， {\ Mathbb z} $（在这里，$ {\ Mathbb f} _2 $是两个发电机上的免费组）。事实证明，这特别适用于任何组$ g $拟合作为无限呈现的小组的短序列的中期，从而将此类组扩展名分类为适当的$ 2 $等值。
In this paper, we consider an equivalence relation within the class of finitely presented discrete groups attending to their asymptotic topology rather than their asymptotic geometry. More precisely, we say that two finitely presented groups $G$ and $H$ are "proper $2$-equivalent" if there exist (equivalently, for all) finite $2$-dimensional CW-complexes $X$ and $Y$, with $π_1(X) \cong G$ and $π_1(Y) \cong H$, so that their universal covers $\widetilde{X}$ and $\widetilde{Y}$ are proper $2$-equivalent. It follows that this relation is coarser than the quasi-isometry relation. We point out that finitely presented groups which are $1$-ended and semistable at infinity are classified, up to proper $2$-equivalence, by their fundamental pro-group, and we study the behaviour of this relation with respect to some of the main constructions in combinatorial group theory. A (finer) similar equivalence relation may also be considered for groups of type $F_n, n \geq 3$, which captures more of the large-scale topology of the group. Finally, we pay special attention to the class of those groups $G$ which admit a finite $2$-dimensional CW-complex $X$ with $π_1(X) \cong G$ and whose universal cover $\widetilde{X}$ has the proper homotopy type of a $3$-manifold. We show that if such a group $G$ is $1$-ended and semistable at infinity then it is proper $2$-equivalent to either ${\mathbb Z} \times {\mathbb Z} \times {\mathbb Z}$, ${\mathbb Z} \times {\mathbb Z}$ or ${\mathbb F}_2 \times {\mathbb Z}$ (here, ${\mathbb F}_2$ is the free group on two generators). As it turns out, this applies in particular to any group $G$ fitting as the middle term of a short exact sequence of infinite finitely presented groups, thus classifying such group extensions up to proper $2$-equivalence.
