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.

Original languageEnglish (US)
Pages (from-to)47-63
Number of pages17
JournalCommentarii Mathematici Helvetici
Issue number1
StatePublished - Mar 2021
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • General Mathematics


  • Atoroidal
  • Group invariant
  • Irreducible
  • Mapping torus


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