Relative hyperbolicity of ascending HNN extension of groups

TL;DR

This paper proves that certain ascending HNN extensions of groups are relatively hyperbolic when the group and endomorphism satisfy specific conditions, extending understanding of hyperbolic structures in group theory.

Contribution

It establishes conditions under which ascending HNN extensions are relatively hyperbolic, especially for free groups with exponentially growing endomorphisms.

Findings

Ascending HNN extensions of groups with preserved free factor systems are relatively hyperbolic.

Exponentially growing endomorphisms of free groups lead to relatively hyperbolic ascending HNN extensions.

Provides a new perspective on the hyperbolic structure of complex group extensions.

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.