Asphericity of cubical presentations: the general case

Abstract

We show that, under suitable hypotheses, the coned-off spaces associated to $C(9)$ cubical small-cancellation presentations are aspherical, and use this to provide classifying spaces, or classifying spaces for proper actions, for their fundamental groups. Along the way, we show that the Cohen--Lyndon property holds for the subgroups of the fundamental group of a non-positively curved cube complex associated to a $C(9)$ cubical presentation, and thus obtain near-sharp upper and lower bounds for the (rational) cohomological dimension of these quotients. We apply these results to give an alternative construction of compact $K(\pi,1)$ for Artin groups with no labels in $\{3,4\}$, from which a new direct sum decomposition for their homology and cohomology with various coefficients above a certain dimension follows. We also address a question of Wise about the virtual torsion-freeness of cubical small-cancellation groups.