Previously, Reynolds showed that any irreducible nonsurjective endomorphism can be represented by an irreducible immersion on a finite graph. We give a new proof of this and also show a partial converse holds when the immersion has connected Whitehead graphs with no cut vertices. The next result is a characterization of finitely generated subgroups of the free group that are invariant under an irreducible nonsurjective endomorphism. Consequently, irreducible nonsurjective endomorphisms are fully irreducible. The characterization and Reynolds' theorem imply that the mapping torus of an irreducible nonsurjective endomorphism is word‐hyperbolic.

中文翻译：

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Fn的不可约非排斥同构是双曲的

以前，雷诺兹（Reynolds）表明，任何不可归约的非排斥性同形可以用有限图上不可归约的沉浸来表示。我们给出了一个新的证明，并且当浸入连接了没有割点的Whitehead图时，还显示了部分相反的成立。下一个结果是对在不可还原的非排斥内同态下不变的有限组的有限生成子组的表征。因此，不可还原的非排斥同构是完全不可还原的。特征和雷诺定理表明，不可归约的非排斥同构同形的映射圆环是双曲线的。