Homological conditions for graphical splittings of antisocial graph groups

We define an antisocial graph group to be a graph group arising from a graph whose clique graph is triangle free. In every dimension, the existence of graphical splittings of an antisocial graph group G is shown to correspond to nonvanishing of Betti numbers of the ball of radius 1 in the well-known cube complex on which G acts freely. This can be regarded as a first step towards generalising Stallingsʹ ends theorem to higher dimensions.