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 FSG be a certain Z2G-module. In this paper, using the cohomological invariant E(G,S,FSG) or simply 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 ``singG(S)" defined by Kropholler and Roller ([8]. We conclude that E~(G,S) can substitute the obstruction ``singG(S)" in more general way. We also analyze splittings of groups in the case, when G and S satisfy certain duality conditions.