论文标题
塔和双曲线群的一阶理论
Towers and the first-order theory of hyperbolic groups
论文作者
论文摘要
本文致力于无扭转双曲线群的一阶理论。它的目的之一是审查一些结果，并提供精确，正确的陈述和定义，以及一些证明和新结果。 一个关键概念是塔（Sela）或NTQ系统（Kharlampovich-Myasnikov）的概念。我们彻底讨论它们。 我们陈述并证明了一种新的一般定理，该定理在文献中统一了几个结果：基本的无扭转双曲线群具有同构内核（SELA）；如果$ h $基本嵌入了无扭转双曲线$ g $中，则$ g $相对于$ h $（perin）的$ h $超过$ h $;自由组（Perin-Sklinos，Ould-Houcine）以及更普遍的原型和自由组的免费产品是同质的。 与Sela和Perin的结果相反是正确的。这是从塔斯基（Tarski）对自由群体基本等效性的问题的解决方案，这是由于Sela和Kharlampovich-Myasnikov而独立的，我们将其视为整个纸张中的黑匣子。 我们提供了许多例子和反例，我们证明了一些新的模型理论结果。我们表征无扭转双曲线组之间的主要模型，而基本自由组之间的模型最小。使用fraïssé的方法，我们将每个无扭转双曲线组$ h $ h $关联到一个唯一的同质可视群$ \ mathcal {m} $，其中任何双曲线组$ h'$等同于$ h $的$ h $等于$ h $。 在附录中，由于Sela，我们给出了一个事实的完整证明，该事实是在无扭转双曲线组$ h $上的塔楼是$ h $ limit groups。
This paper is devoted to the first-order theory of torsion-free hyperbolic groups. One of its purposes is to review some results and to provide precise and correct statements and definitions, as well as some proofs and new results. A key concept is that of a tower (Sela) or NTQ system (Kharlampovich-Myasnikov). We discuss them thoroughly. We state and prove a new general theorem which unifies several results in the literature: elementarily equivalent torsion-free hyperbolic groups have isomorphic cores (Sela); if $H$ is elementarily embedded in a torsion-free hyperbolic group $G$, then $G$ is a tower over $H$ relative to $H$ (Perin); free groups (Perin-Sklinos, Ould-Houcine), and more generally free products of prototypes and free groups, are homogeneous. The converse to Sela and Perin's results just mentioned is true. This follows from the solution to Tarski's problem on elementary equivalence of free groups, due independently to Sela and Kharlampovich-Myasnikov, which we treat as a black box throughout the paper. We present many examples and counterexamples, and we prove some new model-theoretic results. We characterize prime models among torsion-free hyperbolic groups, and minimal models among elementarily free groups. Using Fraïssé's method, we associate to every torsion-free hyperbolic group $H$ a unique homogeneous countable group $\mathcal{M}$ in which any hyperbolic group $H'$ elementarily equivalent to $H$ has an elementary embedding. In an appendix we give a complete proof of the fact, due to Sela, that towers over a torsion-free hyperbolic group $H$ are $H$-limit groups.
