A note about splittings of groups and commensurability under a cohomological point of view

Abstract

Let \(G\) be a group, let \(S\) be a subgroup with infinite index in \(G\) and let \(\mathcal{F}_SG\) be a certain \(\mathbb{Z}_2G\)-module. In this paper, using the cohomological invariant \(E(G, S, \mathcal{F}_SG)\) or simply \(\tilde{E}(G,S)\) (defined in [2]), we analyze some results about splittings of group \(G\)  over a commensurable with \(S\) subgroup which are related with the algebraic obstruction ``\(\operatorname{sing}_G(S)\)" defined by Kropholler and Roller ([8]. We conclude that \(\tilde{E}(G,S)\) can substitute the  obstruction ``\(\operatorname{sing}_G(S)\)" in  more general way. We also analyze  splittings of groups in the case, when \(G\) and \(S\) satisfy certain duality conditions.