# Relative growth in hyperbolic groups

## TL;DR

This paper investigates the relative growth rates of normal subgroups within non-elementary hyperbolic groups, especially those with free abelian quotients, and shows that their growth series are not rational.

## Contribution

It provides new estimates on the relative growth of certain subgroups and demonstrates the non-rationality of their growth series, advancing understanding of hyperbolic group structures.

## Key findings

- Relative growth estimates for normal subgroups
- Non-rationality of the associated growth series
- Special focus on subgroups with free abelian quotients

## Abstract

In this note we obtain estimates on the relative growth of normal subgroups of non-elementary hyperbolic groups, particularly those with free abelian quotient. As a corollary, we deduce that the associated relative growth series fail to be rational.