On residual nilpotence of group extensions

Abstract

We study the following question: under what conditions extension of one residually nilpotent group by another residually nilpotent group is residually nilpotent? We prove some sufficient conditions under which this extension is residually nilpotent. Also, we study this question for semi-direct products and, in particular, for extensions of free group by infinite cyclic group: [Formula: see text]. We find conditions under which this group is residually nilpotent, find conditions under which this group has long lower central series. In particular, we prove that for [Formula: see text] the length of the lower central series of [Formula: see text] is equal to 2, [Formula: see text] or [Formula: see text].