Irreducible nonsurjective endomorphisms of Fn are hyperbolic

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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.

Original languageEnglish (US)
Pages (from-to)960-976
Number of pages17
JournalBulletin of the London Mathematical Society
Issue number5
StatePublished - Oct 1 2020
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • General Mathematics


  • 20E05
  • 20E08
  • 20F65
  • 20F67 (primary)


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