{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,4,1]],"date-time":"2023-04-01T11:51:02Z","timestamp":1680349862031},"reference-count":7,"publisher":"World Scientific Pub Co Pte Lt","issue":"03","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Algebra Comput."],"published-print":{"date-parts":[[2003,6]]},"abstract":"<jats:p> Lyndon's group F<jats:sup>Z<\/jats:sup>[x] is the free exponential group over the ring of integral polynomials Z[x]. This group, introduced by Lyndon in the 1960s, continues to be of interest to group theorists due to its importance in the study of first-order properties of free groups, in particular, equations over free groups. One of the crucial results of Lyndon's study was that the group F<jats:sup>Z<\/jats:sup>[x] is fully residually F; i.e. for any finite collection of nontrivial elements in F<jats:sup>Z<\/jats:sup>[x] there exists a homomorphism \u03c6 : F<jats:sup>Z<\/jats:sup>[x] \u2192 F which is the identity on F and maps the given elements of F<jats:sup>Z<\/jats:sup>[x] into nontrivial elements of F. The importance of F<jats:sup>Z<\/jats:sup>[x] was further emphasized when Kharlampovich and Myasnikov proved in [3] that a finitely generated group is fully residually free if and only if it is embeddable into F<jats:sup>Z<\/jats:sup>[x]. Lyndon's group and its subgroups play a vital role in the technique employed by O. Kharlampovich and A. Myasnikov in their solution of the famous Tarski problem on the decidability of the elementary theory of a free group (see [4, 5]). <\/jats:p><jats:p> In this paper, we show that Lyndon's group is conjugately residually free, i.e. it is possible to map F<jats:sup>Z<\/jats:sup>[x] to the free group F preserving the nonconjugacy of two elements. This result is a further step towards the understanding of the properties of F<jats:sup>Z<\/jats:sup>[x]; moreover, it is closely related to the problem of \"lifting solutions\" of equations from F to F<jats:sup>Z<\/jats:sup>[x], since our result implies that the solutions can indeed be \"lifted\" from F to F<jats:sup>Z<\/jats:sup>[x] for equations of the type x<jats:sup>-1<\/jats:sup> c<jats:sub>1<\/jats:sub> x = c<jats:sub>2<\/jats:sub>. <\/jats:p><jats:p> The structure of Lyndon's group, described by A. Myasnikov and V. Remeslennikov in [8], involves an infinite sequence of free constructions of a specific type, called free extensions of centralizers. For more results on residual properties of certain types of free constructions, see also the works of Ribes, Segal and Zalesskii (for example [9]). <\/jats:p>","DOI":"10.1142\/s0218196703001420","type":"journal-article","created":{"date-parts":[[2003,8,20]],"date-time":"2003-08-20T09:30:13Z","timestamp":1061371813000},"page":"255-275","source":"Crossref","is-referenced-by-count":6,"title":["Lyndon's Group is Conjugately Residually Free"],"prefix":"10.1142","volume":"13","author":[{"given":"Ekaterina","family":"Lioutikova","sequence":"first","affiliation":[{"name":"Department of Mathematical Sciences,  Saint Joseph College, 1678 Asylum Avenue, West Hartford, CT 06117, USA"}]}],"member":"219","published-online":{"date-parts":[[2011,11,20]]},"reference":[{"key":"rf3","doi-asserted-by":"publisher","DOI":"10.1006\/jabr.1997.7184"},{"key":"rf4","volume-title":"Tarski's problem about the elementary theory of free groups has a positive solution","author":"Kharlampovich O.","year":"1998"},{"key":"rf6","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-61896-3"},{"key":"rf7","first-page":"1106","volume":"35","author":"Myasnikov A. G.","journal-title":"Siberian Math. J."},{"key":"rf8","doi-asserted-by":"publisher","DOI":"10.1142\/S0218196796000398"},{"key":"rf9","doi-asserted-by":"publisher","DOI":"10.1112\/S0024610798006267"},{"key":"rf10","doi-asserted-by":"publisher","DOI":"10.2307\/2371129"}],"container-title":["International Journal of Algebra and Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218196703001420","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,6]],"date-time":"2019-08-06T22:25:23Z","timestamp":1565130323000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0218196703001420"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2003,6]]},"references-count":7,"journal-issue":{"issue":"03","published-online":{"date-parts":[[2011,11,20]]},"published-print":{"date-parts":[[2003,6]]}},"alternative-id":["10.1142\/S0218196703001420"],"URL":"https:\/\/doi.org\/10.1142\/s0218196703001420","relation":{},"ISSN":["0218-1967","1793-6500"],"issn-type":[{"value":"0218-1967","type":"print"},{"value":"1793-6500","type":"electronic"}],"subject":[],"published":{"date-parts":[[2003,6]]}}}