# Degrees of compression and inertia for free-abelian times free groups

## TL;DR

This paper introduces new measures called degree of inertia and degree of compression for subgroups in free-abelian times free groups, providing formulas and bounds for these concepts.

## Contribution

It defines and computes the degree of compression and inertia for subgroups in free-abelian times free groups, including explicit formulas and bounds.

## Key findings

- Computed degree of compression for direct products of free-abelian and free groups.
- Provided an upper bound for the degree of inertia in these groups.
- Introduced restricted degree of inertia and related it to projections onto free parts.

## Abstract

We introduce the concepts of degree of inertia, $\text{di}_G(H)$, and degree of compression, $\text{dc}_G(H)$, of a finitely generated subgroup $H$ of a given group $G$. For the case of direct products of free-abelian and free groups, we compute the degree of compression and give an upper bound for the degree of inertia. Imposing some technical assumptions to the supremum involved in the definition of degree of inertia, we introduce the notion called restricted degree of inertia, $\text{di}'_G(H)$, and, again for the case $\mathbb{Z}^m \times F_n$, we provide an explicit formula relating it to the restricted degree of inertia of its projection to the free part, $\text{di}'_{F_n}(H\pi)$.