TY - JOUR
T1 - Endomorphisms, train track maps, and fully irreducible monodromies
AU - Dowdall, Spencer
AU - Kapovich, Ilya
AU - Leininger, Christopher J.
N1 - Funding Information:
1 The author was partially supported by the NSF postdoctoral fellowship, NSF MSPRF no. 1204814. 2 The author was partially supported by the NSF grants DMS-0904200, DMS-1405146, and by the Simons Foundation Collaboration grant no. 279836. 3 The author was partially supported by the NSF grant DMS-1207183 and acknowledges support from NSF grants DMS 1107452, 1107263, 1107367 “RNMS: Geometric structures and representation varieties” (the GEAR Network).
Publisher Copyright:
© 2017 European Mathematical Society.
PY - 2017
Y1 - 2017
N2 - 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.
AB - 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.
KW - Bieri-Neumann-Strebel invariant
KW - Free group endomorphism
KW - Free-by-cyclic group
KW - Fully irreducible automorphism
KW - Train track representative
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U2 - 10.4171/GGD/425
DO - 10.4171/GGD/425
M3 - Article
AN - SCOPUS:85038859299
VL - 11
SP - 1179
EP - 1200
JO - Groups, Geometry, and Dynamics
JF - Groups, Geometry, and Dynamics
SN - 1661-7207
IS - 4
ER -