Title: A topological equivalence relation for finitely presented groups Show Chinese title

Title: 有限表示群的拓扑等价关系

Abstract: 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 $\pi_1(X) \cong G$ and $\pi_1(Y) \cong H$, 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 $\pi_1(X) \cong G$ 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. Show Chinese abstract

Abstract: 在本文中，我们考虑在有限表示的离散群类中的一种等价关系，关注的是它们的渐近拓扑而非渐近几何。 更准确地说，我们说两个有限表示群$G$和$H$是“proper$2$-等价的”，如果存在（等价地，对于所有）有限的$2$-维CW复形$X$和$Y$，具有$\pi_1(X) \cong G$和$\pi_1(Y) \cong H$，使得它们的万有覆盖空间$\widetilde{X}$和$\widetilde{Y}$是 proper$2$-等价的。 因此，这种关系比拟等距关系更粗略。 我们指出，那些在无穷远处是$1$-端且半稳定的有限表示群，可以按照其基本 pro-群，在适当的$2$-等价下被分类，并且我们研究了这种关系在组合群论的一些主要构造中的行为。 对于类型为$F_n, n \geq 3$的群，也可以考虑一种（更细）类似的等价关系，它能捕捉到该群更大的尺度拓扑结构。 最后，我们特别关注那些群 $G$，它们允许一个有限的 $2$-维 CW 复形 $X$，具有 $\pi_1(X) \cong G$，并且其万有覆盖 $\widetilde{X}$具有与 $3$-流形相同的适当同伦类型。 我们证明，如果这样的群$G$是$1$-ended 且在无穷远处是半稳定的话，那么它与${\mathbb Z} \times {\mathbb Z} \times {\mathbb Z}$、${\mathbb Z} \times {\mathbb Z}$或${\mathbb F}_2 \times {\mathbb Z}$（这里，${\mathbb F}_2$是两个生成元的自由群）是$2$-等价的。 正如所显示的，这特别适用于任何作为无限有限表示群的短正合序列中间项的群$G$，从而根据适当的$2$-等价对这样的群扩张进行分类。