Irreducible nonsurjective endomorphisms of free groups are fully irreducible and their mapping tori are word-hyperbolic.
Hyperbolic automorphisms of free groups
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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.