  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,
$\text{di}'_{F_n}(H\pi)$.
