Irreducible Nonsurjective Endomorphisms of $F_n$ are Hyperbolic

TL;DR

This paper proves that irreducible nonsurjective endomorphisms of free groups are hyperbolic, providing new insights into their structure and subgroup invariance, with implications for the hyperbolicity of their mapping tori.

Contribution

It offers a new proof of Reynolds' theorem, characterizes invariant subgroups, and establishes the hyperbolicity of mapping tori for these endomorphisms.

Findings

Irreducible nonsurjective endomorphisms are fully irreducible.

Subgroups invariant under such endomorphisms are characterized.

Mapping tori of these endomorphisms are word-hyperbolic.

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.