Irreducible Nonsurjective Endomorphisms of $F_n$ are 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.
Keywords
Cite
@article{arxiv.1908.08214,
title = {Irreducible Nonsurjective Endomorphisms of $F_n$ are Hyperbolic},
author = {Jean Pierre Mutanguha},
journal= {arXiv preprint arXiv:1908.08214},
year = {2021}
}
Comments
v1: 19 pages, 1 figure. v2: 20 pages, 1 figure. Theorem 4.5 (now 4.6) as previously stated has a counterexample (Example 4.7). We added an extra hypothesis in the statement and adapted the proof accordingly. Consequently, we had to remove the algorithm described at the end of Section 4. Finally, Corollary 4.6 is now Corollary 5.6. v3: Fixed broken references