Groups, Geometry, and Dynamics


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Volume 11, Issue 4, 2017, pp. 1179–1200
DOI: 10.4171/GGD/425

Published online: 2017-12-07

Endomorphisms, train track maps, and fully irreducible monodromies

Spencer Dowdall[1], Ilya Kapovich[2] and Christopher J. Leininger[3]

(1) Vanderbilt University, Nashville, USA
(2) University of Illinois at Urbana-Champaign, USA
(3) University of Illinois at Urbana-Champaign, USA

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 endomorphismof 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 irreducible monodromy 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.

Keywords: Free group endomorphism, train track representative, fully irreducible automorphism, free-by-cyclic group, Bieri-Neumann-Strebel invariant

Dowdall Spencer, Kapovich Ilya, Leininger Christopher: Endomorphisms, train track maps, and fully irreducible monodromies. Groups Geom. Dyn. 11 (2017), 1179-1200. doi: 10.4171/GGD/425