Endomorphisms, train track maps, and fully irreducible monodromies

Spencer Dowdall, Ilya Kapovich, Christopher J. Leininger

Research output: Contribution to journalArticlepeer-review


Any endomorphism of a finitely generated free group naturally descends to an injective endomorphism of its stable quotient. In this paper, we prove a geometric incarnation of this phenomenon: namely, that every expanding irreducible train track map inducing an endomorphism of the fundamental group gives rise to an expanding irreducible train track representative of the injective endomorphism of the stable quotient. As an application,we prove that the property of having fully irreduciblemonodromy for a splitting of a hyperbolic free-by-cyclic group depends only on the component of the BNS-invariant containing the associated homomorphism to the integers.

Original languageEnglish (US)
Pages (from-to)1179-1200
Number of pages22
JournalGroups, Geometry, and Dynamics
Issue number4
StatePublished - 2017


  • Bieri-Neumann-Strebel invariant
  • Free group endomorphism
  • Free-by-cyclic group
  • Fully irreducible automorphism
  • Train track representative

ASJC Scopus subject areas

  • Geometry and Topology
  • Discrete Mathematics and Combinatorics


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