Diophantine geometry over groups VII: The elementary theory of a hyperbolic group
Diophantine geometry over groups VII: The elementary theory of a hyperbolic group
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VII 群上的丢番图几何：双曲群的基本理论
DOI：
10.1112/plms/pdn052
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发表时间：
2009
影响因子：
1.8
通讯作者：
Z. Sela
中科院分区：
文献类型：
--
作者：
Z. Sela
本文将我们关于自由群中方程组的解的集合的结构、这些集合的投影以及定义在自由群上的基本集合的结构的工作推广到一般的无挠（Gromov）双曲群。特别地，我们证明了这样一个群上的每个可定义集都在AE集生成的布尔代数中，证明了双曲性是AE生成群的一阶不变量，并得到了无挠双曲群的初等等价类的分类.最后，我们给出了一个有效的方法来判定两个给定的无挠双曲群是否初等等价。
This paper generalizes our work on the structure of sets of solutions to systems of equations in a free group, projections of such sets, and the structure of elementary sets defined over a free group, to a general torsion‐free (Gromov) hyperbolic group. In particular, we show that every definable set over such a group is in the Boolean algebra generated by AE sets, prove that hyperbolicity is a first‐order invariant of a finitely generated group, and obtain a classification of the elementary equivalence classes of torsion‐free hyperbolic groups. Finally, we present an effective procedure to decide if two given torsion‐free hyperbolic groups are elementarily equivalent.