Parametrizing Ω ₁(Z): Singular Case

Abstract

In this chapter we work under the blanket assumption that G is a finitely generated group with abelianization G ^ab and integral group ring Λ=Z[G], and that $\operatorname{Ext}_{\varLambda}^{1}(\mathbf{ Z}, \varLambda)\neq0$. Then G is necessarily infinite. We investigate minimality of $\mathcal{I}$ in Ω ₁(Z) and first establish:

Second minimality criterion: :

$\mathcal{I}$ lies at the minimal level of Ω ₁(Z) if G ^ab is finite.

This second criterion also applies to many cases where $\operatorname {Ext}_{\varLambda}^{1}(\mathbf{ Z}, \varLambda) = 0$ although we do not need to use it there. We employ it in Sect. 14.3 to give examples of groups G with infinite splitting in Ω ₁(Z).