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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]. <\/jats:p>","DOI":"10.1142\/s0218196723500613","type":"journal-article","created":{"date-parts":[[2023,8,19]],"date-time":"2023-08-19T05:28:20Z","timestamp":1692422900000},"page":"1617-1650","source":"Crossref","is-referenced-by-count":0,"title":["On residual nilpotence of group extensions"],"prefix":"10.1142","volume":"33","author":[{"given":"V.\u00a0G.","family":"Bardakov","sequence":"first","affiliation":[{"name":"Sobolev Institute of Mathematics, Prosp. 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