# Rank and Nielsen equivalence in hyperbolic extensions

## TL;DR

This paper extends a theorem relating to rank and Nielsen equivalence from hyperbolic fibered 3-manifolds to a broader class of hyperbolic group extensions, including surface groups and free groups.

## Contribution

It generalizes Souto's theorem to a large class of hyperbolic group extensions, broadening its applicability.

## Key findings

- Generalization of Souto's theorem to hyperbolic extensions
- Includes all hyperbolic extensions of surface groups
- Covers hyperbolic extensions of free groups by convex cocompact subgroups

## Abstract

In this note, we generalize a theorem of Juan Souto on rank and Nielsen equivalence in the fundamental group of a hyperbolic fibered 3-manifold to a large class of hyperbolic group extensions. This includes all hyperbolic extensions of surfaces groups as well as hyperbolic extensions of free groups by convex cocompact subgroups of Out$(F_n)$.