We show that in general for a given group the structure of a maximal hyperbolic tower over a free group is not canonical: we construct examples of groups having hyperbolic tower structures over free subgroups which have arbitrarily large ratios between their ranks. These groups have the same first order theory as non-abelian free groups and we use them to study the weight of types in this theory.

中文翻译：

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自由群论中的最大双曲塔和权重

我们表明，一般而言，对于给定的群，自由群上的最大双曲塔结构不是规范的：我们构建了在自由子群上具有双曲塔结构的群的示例，这些自由子群的等级之间具有任意大的比率。这些群具有与非阿贝尔自由群相同的一阶理论，我们使用它们来研究该理论中类型的权重。