Quasirationality and Aspherical (Pro-$p$-) Presentations

It is shown that the property of quasirationality of a (pro-$p$-) presentation of a (pro-$p$-) group $G$ is a property of the (pro-$p$-) group itself and does not depend on the choice of a presentation. It is proved that the class of quasirational presentations is wider than the class of aspherical pro-$p$-presentations (and of combinatorially aspherical presentations in the discrete case). For quasirational presentations, the notion of generalized permutationality of the module of relations is introduced, which turns out to be equivalent to the permutationality of the $\operatorname{mod}(p)$ quotient of the module.