Lower central and dimension series of groups. (Q950927)

The authors of this book present an exposition of different methods related to the theory of the lower central series of groups, the dimension subgroups, the augmentation powers and the derived series. Let \(G\) be a group, \(M\) and \(K\) be subsets of \(G\) and \([H,K]\) be the subgroup of \(G\) generated by all commutators \([h,k]=h^{-1}k^{-1}hk\), \(h\in H\), \(k\in K\). If \(\alpha\) is an ordinal, then the lower central series \(\{\gamma_\alpha(G)\}\) is defined inductively in the following way: \(\gamma_1(G)=G\), \(\gamma_{\alpha+1}(G)=[G,\gamma_\alpha(G)]\) and if \(\alpha\) is a limit ordinal, then \(\gamma_\alpha(G)]=\bigcap_{\beta<\alpha}\gamma_\beta(G)\). A group \(G\) is said to be residually nilpotent if \(\gamma_\omega(G)=1\), where \(\omega\) is the least infinite ordinal. Let \(\Delta_R(G)\) be the augmentation ideal of the group ring \(RG\) of a group \(G\) over a ring \(R\). The augmentation powers \(\Delta_R^n(G)\) are considered. The cross-section \(G\cap(1+\Delta_R^n(G))\), \(n\geq 1\), is a normal subgroup of \(G\). This subgroup is called the \(n\)-th dimension subgroup of \(G\) over \(R\) and is denoted by \(D_{n,R}(G)\). If \(R=\mathbb{Z}\), where \(\mathbb{Z}\) is the ring of integers, then \(D_{n,R}(G)\) is denoted by \(D_n(G)\). The terms of the derived series are defined inductively by setting \(\delta_0(G)=G\), \(\delta_{n+1}(G)=[\delta_n(G),\delta_n(G)]\) for \(n\geq 0\). In Chapter 1 the authors present various constructions and methods for studying the residual nilpotence of groups and investigate the transfinite terms of the lower central series. In this way they do an attempt to develop a ``limit'' theory for lower central series. The question whether \(\gamma_n(G)=\delta_n(G)\) is the subject of intensive investigation over the last fifty years. It is well known that \(D_n(G)=\gamma_n(G)\) for \(n=1\), 2, 3 and that for \(n\geq 4\) this equality is not fulfilled. In Chapter 2 the authors mark the results of \textit{I. B. S. Passi} [Group rings and their augmentation ideals, Lect. Notes Math. 715 (1979; Zbl 0405.20007)] and \textit{N. Gupta} [Free group rings. Contemp. Math. 66. Providence, AMS (1987; Zbl 0641.20022)] and regard the fourth and the fifth dimension subgroups. They recall the description of the fifth dimension subgroup due to Ken-Ichi Tahara (Theorem 2.29) and give a shorter proof of one of his theorems. In Chapter 3 the authors give a method, due to \textit{N. Gupta} and \textit{I. B. S. Passi} [Int. J. Algebra Comput. 17, No. 5-6, 1021-1031 (2007; Zbl 1187.20019)] for studying the derived series of nilpotent groups. Their main aim is the study of the transfinite terms of the derived series of groups. They give an account of the homotopical approach of \textit{R. Strebel} [Comment. Math. Helv. 49, 302-332 (1974; Zbl 0288.20066)]. The asphericity conjecture of \textit{J. H. C. Whitehead} [Ann. Math. (2) 42, 409-428 (1941; Zbl 0027.26404)] is also explored. In Chapter 4 the authors discuss certain (co)homological methods for the study of group rings, in particular the augmentation powers. In Chapter 5 the authors develop a connection between lower central and dimension series of groups, simplicial homotopy theory and derived functors of non-additive functors in the sense of Dold-Puppe. The spectral sequence investigated by E. Curtis plays a basic role in this study. In Chapter 6 assorted examples involving a group ring construction are presented. In the Appendix the authors give various notions and the tools required from simplicial homotopy theory. This book is very useful for future investigations of the lower central series of groups and the dimension subgroups.