Relative hyperbolicity of hyperbolic-by-cyclic groups

TL;DR

This paper proves that the mapping torus of a torsion-free hyperbolic group by an automorphism is relatively hyperbolic, with the peripheral subgroups being suspensions of polynomially growing subgroups, revealing structural insights into such groups.

Contribution

It introduces a canonical collection of polynomially growing subgroups and establishes the relative hyperbolicity of the mapping torus with respect to their suspensions.

Findings

The existence of a canonical collection of polynomially growing subgroups.

The mapping torus is hyperbolic relative to suspensions of these subgroups.

Structural characterization of automorphisms of hyperbolic groups.

Abstract

Let $G$ be a torsion-free hyperbolic group and $\alpha$ an automorphism of $G$. We show that there exists a canonical collection of subgroups that are polynomially growing under $\alpha$, and that the mapping torus of $G$ by $\alpha$ is hyperbolic relative to the suspensions of the maximal polynomially growing subgroups under $\alpha$.