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

Abstract

We provide an explicit computation of the cohomology groups (with untwisted coefficients) of semidirect products of the form ${\mathbb{Z}}^n⋊\mathbb{Z}/m$ 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 Lück-Davis and Adem-Ge-Pan-Petrosyan. In order to show that our formulas are usable, we develop a concrete example of the form ${\mathbb{Z}}^5⋊\mathbb{Z}/6$ where its cohomology groups are described in full detail.

22 pages, major revision