On the group cohomology of the semi-direct product Z^n rtimes Z/m and a conjecture of Adem-Ge-Pan-Petrosyan

TL;DR

This paper investigates the group cohomology of semi-direct products Z^n rtimes Z/m, proving a conjecture under certain conditions and providing a counterexample in the general case, advancing understanding of spectral sequence behavior.

Contribution

It proves a conjecture about spectral sequence collapse for specific actions and provides a counterexample showing the conjecture does not hold universally.

Findings

Spectral sequence collapses when Z/m acts freely outside the origin.

Counterexample with n=6, m=4 where the second differential is non-zero.

Advances understanding of cohomology in semi-direct product groups.

Abstract

Consider the semi-direct product Z^n rtimes Z/m. A conjecture of Adem-Ge-Pan-Petrosyan predicts that the associated Lyndon-Hochschild-Serre spectral sequence collapses. We prove this conjecture provided that the Z/m-action on Z^n is free outside the origin. We disprove the conjecture in general, namely, we give an example with n=6 and m=4, where the second differential does not vanish.