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Irreducibility of a Free Group Endomorphism is a Mapping Torus Invariant
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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.
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Irreducible Nonsurjective Endomorphisms of $F_n$ are Hyperbolic
Irreducible nonsurjective endomorphisms of free groups are fully irreducible and their mapping tori are word-hyperbolic.
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