Endomorphisms, train track maps, and fully irreducible monodromies

TL;DR

This paper establishes a geometric framework linking endomorphisms of free groups to train track maps, showing how properties of monodromies depend on specific invariants, with implications for hyperbolic free-by-cyclic groups.

Contribution

It provides a geometric interpretation of endomorphisms via train track maps and relates monodromy irreducibility to the BNS-invariant in hyperbolic free-by-cyclic groups.

Findings

Every expanding irreducible train track map induces a corresponding map on the stable quotient.

The property of fully irreducible monodromy depends only on the BNS-invariant component.

Application to hyperbolic free-by-cyclic groups shows invariance of monodromy properties.

Abstract

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 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.