Irreducibility of a Free Group Endomorphism is a Mapping Torus Invariant. [submitted]
Abstract: 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. (pdf, arxiv)
Irreducible Nonsurjective Endomorphisms of \(F_n\) are Hyperbolic. [submitted]
Abstract: Previously, Reynolds showed that any irreducible nonsurjective endomorphism can be represented by an irreducible immersion on a finite graph. We give a new proof of this and also show a partial converse holds when the immersion has connected Whitehead graphs with no cut vertices. The next result is a characterization of finitely generated subgroups of the free group that are invariant under an irreducible nonsurjective endomorphism. Consequently, irreducible nonsurjective endomorphisms are fully irreducible. The characterization and Reynolds' theorem imply that the mapping torus of an irreducible nonsurjective endomorphism is word-hyperbolic. (pdf, arxiv)
Hyperbolic Immersions of Free Groups. to appear in Groups Geom. Dyn.
Abstract: We prove that the mapping torus of a graph immersion has a word-hyperbolic fundamental group if and only if the corresponding endomorphism does not produce Baumslag-Solitar subgroups. Due to a result by Reynolds, this theorem applies to all injective endomorphisms of \(F_2\) and nonsurjective fully irreducible endomorphisms of \(F_n\). We also give a framework for extending the theorem to all injective endomorphisms of \(F_n\). (pdf, arxiv)
Last updated: Sat 25 January 2020