Cut points and canonical splittings of hyperbolic groups
Cut points and canonical splittings of hyperbolic groups
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DOI：
10.1007/bf02392898
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发表时间：
1998-09
期刊：
影响因子：
3.7
通讯作者：
B. Bowditch
中科院分区：
文献类型：
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作者：
B. Bowditch
本文利用边界的局部割点结构，构造了单端双曲群（Gromov [Gr]意义下）的JSJ分裂.特别地，这给出了分裂的拟等距不变性，以及双曲群的环带定理。分裂的正则性质也直接来自于这种方法。JSJ分裂的概念，在这种情况下，由Sela [Se]引入，他为所有（无挠）双曲群构造了这样的分裂。它们的名字来自于与雅科和沙伦[JS]和约翰森[Jo]描述的不可约3-流形的特征子流形构造的类比（发展了Waldhausen早先概述的理论）。JSJ分裂给出了群在两端子群上的所有可能分裂的集合的描述，从而告诉我们外自同构群的结构。我们将假设在这里的事实，即边界是局部连接，即”皮亚诺连续“！这是已知的情况下，所有的单端双曲群，从结果[Bol]，[Bo 2]，[L]，[Sw]，[Bo 5]，我们将很快讨论。这使用了这样一个事实，即局部连通性是由全局截点的不存在所暗示的。
In this paper, we give a construction of the JSJ splitting of a one-ended hyperbolic group (in the sense of Gromov [Gr]), using the local cut point structure of the boundary. In particular, this gives the quasiisometry invariance of the splitting, as well as the annulus theorem for hyperbolic groups. The canonical nature of the splitting is also immediate from this approach.The notion of a JSJ splitting, in this context, was introduced by Sela [Se], who constructed such splittings for all (torsion-free) hyperbolic groups. They take their name from the analogy with the characteristic submanifold construction for irreducible 3-manifolds described by Jaco and Shalen [JS] and Johannson [Jo](developing a theory outlined earlier by Waldhausen). The JSJ splitting gives a description of the set of all possible splittings of the group over two-ended subgroups, and thus tells us about the structure of the outer automorphism group. We shall take as hypothesis here the fact that the boundary is locally connected, ie a" Peano continuum'!. This is now known to be the case for all one-ended hyperbolic groups, from the results of [Bol],[Bo2],[L],[Sw],[Bo5], as we shall discuss shortly. This uses the fact that local connectedness is implied by the non-existence of a global cut point [BM].