Limit groups for relatively hyperbolic groups. I. The basic tools
Limit groups for relatively hyperbolic groups. I. The basic tools
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相对双曲群的极限群。
DOI：
10.2140/agt.2009.9.1423
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发表时间：
2004
影响因子：
0.7
通讯作者：
D. Groves
中科院分区：
文献类型：
--
作者：
D. Groves
我们开始研究-极限群，其中是一个相对于自由阿贝尔子群的集合是双曲的无扭群。利用式（16）的结果，我们将式（21）和式（22）的结果适应于此背景。具体地说，给定一个有限生成群G和一对非共轭半纯态序列{hn: G ！}，我们提取了一个具有非平凡等距g作用的r树。这一点，连同Sela的缩短论证的类比，使我们能够证明本文的主要结果，即Hopfian。在他著名的系列论文（38,39,41）中，Z. Sela将那些有限生成的群与秩2的自由群用相同的基本理论进行了分类（也见（40）作为摘要）。这类包括所有非贝尔自由群、大多数曲面群和某些其他双曲群。特别是，Sela积极地回答了Tarski的一些长期存在的问题（Kharlampovich和Miasnikov对这些问题有另一种方法；见（29））。在（38）中，Sela从极限群的研究开始。Sela对极限群的定义是几何的，尽管它证明一个群是极限群当且仅当它是有限生成的全残馀自由群。Sela随后产生了Makanin-Razborov图，给出了homg （G，F）的参数化，其中G是任意有限生成的群，F是非abel自由群（这种参数化也在（28）中给出）。在他的六篇论文中，Sela使用的两个主要工具是r树上的等距作用理论和缩短论证。Sela的工作自然提出了一个问题，即哪些其他类别的群体可以使用Sela的方法来理解。塞拉的许多方法（更引人注目的是，一些答案）来自几何群论。因此，在寻找将这些方法应用于几何群论的群的类别时，考虑感兴趣的群似乎是很自然的。在（42）中，Sela考虑了任意的无扭转
We begin the investigation of -limit groups, where is a torsion-free group which is hyperbolic relative to a collection of free abelian subgroups. Using the results of (16), we adapt the re- sults from (21) and (22) to this context. Specifically, given a finitely generated group G, and a sequence of pairwise non-conjugate ho- momorphisms {hn : G ! }, we extract an R-tree with a nontrivial isometric G-action. This, along with the analogue of Sela's shortening argument allows us to prove the main result of this paper, that is Hopfian. In his remarkable series of papers (38, 39, 41), Z. Sela has classified those finitely generated groups with the same elementary theory as the free group of rank 2 (see also (40) for a summary). This class includes all nonabelian free groups, most surface groups, and certain other hyperbolic groups. In particular, Sela answers in the positive some long-standing questions of Tarski (Kharlampovich and Miasnikov have another approach to these problems; see (29)). In (38), Sela begins with a study of limit groups. Sela's definition of a limit group is geometric, though it turns out that a group is a limit group if and only if it is a finitely generated fully-residually free group. Sela then produces Makanin-Razborov diagrams, which give a parametrization of Hom(G,F), where G is an arbitrary finitely gener- ated group and F is a nonabelian free group (such a parametrisation is also given in (28)). Over the course of his six papers, two of the main tools Sela uses are the theory of isometric actions on R-trees and the shortening argument. Sela's work naturally raises the question of which other classes of groups can be understood using Sela's approach. Many of Sela's meth- ods (and, more strikingly, some of the answers) come from geometric group theory. Thus it seems natural to consider, when looking for classes of groups to apply these methods to, groups of interest in geo- metric group theory. In (42), Sela considers an arbitrary torsion-free