We prove that the mapping torus of a graph immersion has a word-hyperbolic fundamental group if and only if the corresponding endomorphism does not produce Baumslag–Solitar subgroups. Due to a result by Reynolds, this theorem applies to all injective endomorphisms of $F_2$ and nonsurjective fully irreducible endomorphisms of $F_n$. We also give a framework for extending the theorem to all injective endomorphisms of $F_n$.

中文翻译：

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自由群体的双曲线沉浸

我们证明，当且仅当相应的内同态不产生Baumslag–Solitar子组时，图浸入的映射圆环才具有词双曲基本组。由于雷诺兹的结果，该定理适用于$ F_2 $的所有内射内同态和$ F_n $的非外射完全不可约的内同态。我们还提供了一个框架，将定理扩展到$ F_n $的所有内射内同态。