Homology of the pronilpotent completion and cotorsion groups (Q6588716)

Let \(F\) be a non-cyclic free group and \(\widehat{F}\) denote its pronilpotent completion: the inverse limit of the quotients \(F/\gamma_i(F)\), where \(\{\gamma_i(F)\}\) denotes the lower central series of \(F\). The (complicated) structure of \(\widehat{F}\) is of interest both inherently and for the more general study of parafree groups (groups that are residually nilpotent and have the same lower central quotients as a free group). The authors discuss several open homological questions about \(\widehat{F}\) and conjecture homological conditions for a residually nilpotent group to be parafree. The focus of this paper is the second integral homology group \(H_2(\widehat{F},\mathbb{Z})\) that was previously known to be uncountable. The main result is that \(H_2(\widehat{F},\mathbb{Z})\) is \textit{not} a cotorsion group (a group having no non-trivial extension with the group of rational numbers). Were the second cohomology cotorsion, then we would have \(\operatorname{Hom}(H_2(\widehat{F},\mathbb{Z}),\mathbb{Z}) = 0\); something the authors still conjecture to hold.\N\N\NThe proof makes use of an epimorphism \(F \twoheadrightarrow L\), for the integral lamplighter group \(L\), along with two significant intermediate results. Using a generalization of a formula of Hopf, for an arbitrary group epimorphism \(G \to Q\) (with \(G\) having finitely generated abelianization), it is shown that the cokernel of the map \(H_2(\widehat{G},\mathbb{Z})\oplus H_2(Q,\mathbb{Z}) \to H_2(\widehat{Q},\mathbb{Z})\) is cotorsion. On the other hand, it is shown that the cokernel of the map \(H_2(L,\mathbb{Z}) \to H_2(\widehat{L},\mathbb{Z})\) is not cotorsion. This latter result follows from a non-trivial technical result about certain power series groups not being cotorsion. Some further independent results in that direction are also provided.\N\NIn addition, the authors show how their methodology may be used to get analogous results for a free integral Lie algebra having more than one generator.