We prove that the property of a free group endomorphism being irreducible is a group invariant of the ascending HNN extension it defines. This answers a question posed by Dowdall–Kapovich–Leininger. We further prove that being irreducible and atoroidal is a commensurability invariant. The invariance follows from an algebraic characterization of ascending HNN extensions that determines exactly when their defining endomorphisms are irreducible and atoroidal; specifically, we show that the endomorphism is irreducible and atoroidal if and only if the ascending HNN extension has no infinite index subgroups that are ascending HNN extensions.

中文翻译：

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游离基团同型的不可约性是一个映射环面不变性

我们证明，不可还原的自由基团同质性的性质是其定义的递增HNN扩展的基团不变性。这回答了道达尔-卡波维奇-莱宁格提出的一个问题。我们进一步证明，不可约和有齿是不变的。不变性来自于递增的HNN扩展的代数特征，该特征精确地确定了何时它们定义的内同态是不可约的和有代数的；具体而言，我们表明，当且仅当升序HNN扩展不具有作为升序HNN扩展的无限索引子组时，内同态是不可约的且是无齿的。