Groups with the same lower central sequence as a relatively free group. II. Properties

Abstract

0.0. This paper is a continuation of [3], which contains the motivation for this work. Many of the terms and notations used in [3] will be used throughout this paper without any explicit reference. Suppose that 3 is a variety of groups. A group P is termed parafree(2) in 3, or a parafree 3-group, or simply parafree if there is no question as to which variety is involved, if (i) P is a 3-group i.e., P ce 3; (ii) P is residually nilpotent; (iii) P has the same lower central sequence(2) as some free 3-group. The object of this paper is to investigate the properties of such parafree groups with the basic hope (which is amply fulfilled) that many properties of the relevant free group persist in the corresponding parafree group. 0.1. The first section is preliminary in nature being essentially elementary. First we prove that if P is a parafree 3-group of finite rank and if N is a normal subgroup of P whose quotient P/N is again parafree in 3 of the same rank as P, then N= 1. This almost obvious fact implies that parafree groups of finite rank are hopfian. Of course parafree groups of rank one are cyclic. It follows easily from this that a parafree group of rank two is freely decomposable if and only if it is free. Next we consider absolutely parafree groups i.e., groups parafree in the variety of all groups. For such groups we prove that free products and free factors are again parafree. 0.2. In ?2 we complement the results in [3] by obtaining some more, rather different, parafree groups. We first consider one-relator groups.