On the group cohomology of groups of the form ℤⁿ\rtimes ℤ/m with m square-free

2024/03/21 by Saldaña, Luis Jorge Sánchez, Velásquez, Mario
#20H15 #20J06 #20K25 #55T25 #Algebraic Topology (math.AT) #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2403.14569

Abstract

We provide an explicit computation of the cohomology groups (with untwisted coefficients) of semidirect products of the form ℤⁿ\rtimes ℤ/m with m free of squares, by means of formulas that only depend on n, m and the action of ℤ/m on ℤⁿ. We want to highlight the fact that we are not impossing any conditions on the ℤ/m-action on ℤⁿ, 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 ℤ⁵\rtimes ℤ/6 where its cohomology groups are described in full detail.

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