The combinatorial structure of cocompact discrete hyperbolic groups
The combinatorial structure of cocompact discrete hyperbolic groups
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DOI：
10.1007/bf00146825
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发表时间：
1984-06
影响因子：
0.5
通讯作者：
J. Cannon
中科院分区：
文献类型：
--
作者：
J. Cannon
组合群论开始于Dehn的研究[7]-[9]的基本组的封闭2维流形承认双曲结构。起初德恩自由使用的参数从双曲几何，但迅速他和其他移动的方向的结果可以说明和证明纯粹的组合和代数方面。瑟斯顿（参见[20]，[21]）最近证明了拓扑学家感兴趣的大类群，虽然显然不适合用组合群论的标准方法进行攻击，但仍然是离散双曲群。他的结果表明，价值返回到几何考虑组合群论。我们将表明，德恩的主要组合定理，特别是他的解决方案的世界和共轭问题的双曲曲面群有简单的几何reinterpretation，这些解决方案，重新解释，是真实的所有cocompact，离散双曲群。我们还将证明，这类群的全局组合结构是特别简单的，因为它们的Cayley群图（Dehn Gruppenbilder）（见[4]-[9]）具有线性递归的描述。我们认为后一个结果表明了小相消理论的一个有希望的推广（见[14]，第二章）。其中小的抵消假设可能局部失败，但在某种意义上，全局成立。这个结果还表明，上紧离散双曲群可以像理解整数Z那样被整体地理解：感觉到，正如我们所做的那样，我们理解了Z中的简单线性递归n~ n+ 1，我们将Z的局部图像递归地扩展到无限远。人们以同样的方式获得了任意余紧离散双曲群G的全局图像：首先，人们发现了G的局部图像，然后发现了G的递归结构，通过递归结构，局部结构的副本被整合。本文的研究结果需要补充具体的COM-
Combinatorial group theory began with Dehn's study [7]-[9] of the fundamental group of the closed 2-dimensional manifold admitting a hyperbolic structure. At first Dehn freely used arguments from hyperbolic geometry, but rapidly he and others moved in the direction of results which could be stated and proved in purely combinatorial and algebraic terms. Thurston (see [20],[21]) has recently shown that large classes of groups of interest to topologists, while not obviously amenable to attack by standard methods of combinatorial group theory, nevertheless are discrete hyperbolic groups. His result suggests the value of a return to geometric considerations in combinatorial group theory.We shall show that Dehn's principal combinatorial theorems-in particular his solutions to the world and conjugacy problems for hyperbolic surface groups-have simple geometric reinterpretations, and that these solutions, as reinterpreted, are true for all cocompact, discrete hyperbolic groups. We shall also show that the global combinatorial structure of such groups is particularly simple in the sense that their Cayley group graphs (Dehn Gruppenbilder)(see [4]-[9]) have descriptions by linear recursion. We view this latter result as indicating a promising generalization of small cancellation theory (see [14, Chap. V]) where small cancellation hypotheses may fail locally but, in some sense, hold globally. The result also indicates that cocompact, discrete hyperbolic groups can be understood globally in the same sense that the integers Z can be under-stood: feeling, as we do, that we understand the simple linear recursion n~ n+ 1 in Z, we extend our local picture of Z recursively in our mind's eye toward infinity. One obtains a global picture of the arbitrary cocompact, discrete hyperbolic group G in the same way: first, one discovers the local picture of G, then the recursive structure of G by means of which copies of the local structure are integrated. The results of this paper need to be complemented by specific com-