On the group cohomology of groups of the form $\mathbb{Z}^n\rtimes \mathbb{Z}/m$ with $m$ square-free

Abstract

We provide an explicit computation of the cohomology groups (with untwisted coefficients) of semidirect products of the form $\mathbb{Z}^n\rtimes \mathbb{Z}/m$ with $m$ free of squares, by means of formulas that only depend on $n$, $m$ and the action of $\mathbb{Z}/m$ on $\mathbb{Z}^n$. We want to highlight the fact that we are not impossing any conditions on the $\mathbb{Z}/m$-action on $\mathbb{Z}^n$, and as far as we know our formulas are the first in the literature in this generality. This generalizes previous computations of L\"uck-Davis and Adem-Ge-Pan-Petrosyan. In order to show that our formulas are usable, we develop a concrete example of the form $\mathbb{Z}^5\rtimes \mathbb{Z}/6$ where its cohomology groups are described in full detail.