Hyperbolicity of Multiple Ascending HNN Extensions of Free Groups

TL;DR

This paper extends the understanding of hyperbolic properties of multiple ascending HNN extensions of free groups to include hyperbolic non-surjective endomorphisms, generalizing prior results on automorphisms.

Contribution

It generalizes previous results by showing hyperbolicity for multiple HNN extensions with hyperbolic non-surjective endomorphisms of free groups.

Findings

Extension of hyperbolicity results to non-surjective endomorphisms

Construction of multiple HNN extensions with hyperbolic properties

Generalization of Bestvina-Feighn-Handel theorem

Abstract

Bestvina-Feighn-Handel show that for finitely many generic and independent hyperbolic automorphisms $\phi_1, \cdots, \phi_r$ of $F_n$, the resulting extension $F_n \rtimes F_r$ is hyperbolic. This paper generalizes the above statement to the case where $\phi_1, \cdots, \phi_r$ are hyperbolic non-surjective endomorphisms of $F_n$. In our case the output is a multiple HNN extension associated to a graph with one vertex and $r$ edges. All edge and vertex groups are isomorphic to $F_n$.