Relative hyperbolicity of ascending HNN extension of groups

Abstract

We prove that for a finitely generated group G with a free factor system and an injective endomorphism that preserves the free factor system, the ascending HNN extension of G is hyperbolic relative to a collection of maximal parabolic subgroups. As a corollary, we see that if an injective endomorphism of a finite rank free group F is exponentially growing, the ascending HNN extension of F is relatively hyperbolic.