On a Theorem of Scott and Swarup

TL;DR

This paper proves that in a hyperbolic group extension induced by a fully irreducible automorphism, any finitely generated distorted subgroup must have finite index, extending a classical surface subgroup theorem to hyperbolic groups.

Contribution

It establishes a new analog of Scott and Swarup's theorem for hyperbolic groups, showing finitely generated distorted subgroups are of finite index.

Findings

Finitely generated distorted subgroups have finite index in the hyperbolic group.

Extension induced by a fully irreducible automorphism preserves subgroup finiteness properties.

Generalizes surface subgroup theorems to hyperbolic group contexts.

Abstract

Let $1 \rightarrow H \rightarrow G \rightarrow \mathbb{Z} \rightarrow 1$ be an exact sequence of hyperbolic groups induced by a fully irreducible automorphism $\phi$ of the free group $H$. Let $H_1 (\subset H)$ be a finitely generated distorted subgroup of $G$. Then $H_1$ is of finite index in $H$. This is an analog of a Theorem of Scott and Swarup for surfaces in hyperbolic 3-manifolds.