Subgroups with projective Abelianization and trivial multiplicator (Q797691)

Let X be a connected subcomplex of a connected two-dimensional, aspherical CW-complex Y and let i:\(X\to Y\) denote the inclusion. Set \(L=\ker_{\pi_ 1}(i)\), put \(G=\pi_ 1(X)\) and \(H=im \pi_ 1(i)\). The resulting extension of groups \(L\hookrightarrow G\twoheadrightarrow H\) is known to have the following two properties: (i) \(H_ 1(L,{\mathbb{Z}})=L/[L,L]\) is a projective \({\mathbb{Z}}H\)-module, (ii) \(H_ 2(L,{\mathbb{Z}})\) is trivial. This example motivates the study of extensions \(L\hookrightarrow G\twoheadrightarrow H\) with (i), (ii) undertaken in the note under review. The main result is Theorem 2.1. If \(L\hookrightarrow G\twoheadrightarrow H\) satisfies (i), (ii) then each quotient \(L_ j/L_{j+1}\) of successive terms of the lower central series of L is a submodule of a free \({\mathbb{Z}}H\)-module. As an application of this result the author proves that, in case H is torsion-free, no element of \(G\backslash L\) centralizes an element of \(L\backslash L_{\omega}\), where \(L_{\omega}=\cap_{j<\omega}L_ j\).