{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,3,30]],"date-time":"2022-03-30T22:55:18Z","timestamp":1648680918970},"reference-count":11,"publisher":"World Scientific Pub Co Pte Lt","issue":"02","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Algebra Comput."],"published-print":{"date-parts":[[2003,4]]},"abstract":"<jats:p> Let F be the free group of rank 2, generated by x and y, and let w(x, y) \u2208 F\u2032 be a non-trivial word. We give elementary algebraic proofs and algorithms to (1) express [x, y]<jats:sup>n<\/jats:sup> as a product of [n\/2] + 1 commutators and show this is the best possible; (2) show that (w(x, y))<jats:sup>2<\/jats:sup> cannot be written as one w-word and if g \u2260 1 \u2208 w(F) then show that the minimal number of w-words required to express g<jats:sup>n<\/jats:sup> as their product tends to infinity with n. Other results for free groups of higher rank are also presented. <\/jats:p>","DOI":"10.1142\/s0218196703001316","type":"journal-article","created":{"date-parts":[[2003,4,17]],"date-time":"2003-04-17T10:15:25Z","timestamp":1050574525000},"page":"231-240","source":"Crossref","is-referenced-by-count":0,"title":["PRODUCTS OF COMMUTATORS IN FREE GROUPS"],"prefix":"10.1142","volume":"13","author":[{"given":"MEHRI","family":"AKHAVAN-MALAYERI","sequence":"first","affiliation":[{"name":"Az-zahra University, Vanak,  Tehran 19834, Iran"}]},{"given":"AKBAR","family":"RHEMTULLA","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences,  University of Alberta, Edmonton, Alberta, Canada T6G 2G1, Canada"}]}],"member":"219","published-online":{"date-parts":[[2011,11,20]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.1007\/BF02546390"},{"key":"rf2","first-page":"109","volume":"37","author":"Bavard C.","journal-title":"Enseign. Math."},{"key":"rf3","first-page":"53","volume":"91","author":"Brooks R.","journal-title":"Riemmanian Surfaces and Related Topics, Annals of Mathematics Study"},{"key":"rf4","doi-asserted-by":"publisher","DOI":"10.1142\/S0218196794000105"},{"key":"rf5","doi-asserted-by":"publisher","DOI":"10.1016\/0040-9383(81)90033-1"},{"key":"rf6","doi-asserted-by":"publisher","DOI":"10.1017\/S0305004100029212"},{"key":"rf7","doi-asserted-by":"publisher","DOI":"10.1017\/S0305004100043231"},{"key":"rf8","doi-asserted-by":"publisher","DOI":"10.4153\/CJM-1969-126-4"},{"key":"rf9","first-page":"60","volume":"21","author":"Romankov V. A.","journal-title":"Algebra i logika"},{"key":"rf10","doi-asserted-by":"publisher","DOI":"10.1112\/S002461150001234X"},{"key":"rf11","first-page":"129","volume":"28","author":"Stewart I.","journal-title":"Proc. London Math. Soc."}],"container-title":["International Journal of Algebra and Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218196703001316","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,6]],"date-time":"2019-08-06T21:56:12Z","timestamp":1565128572000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0218196703001316"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2003,4]]},"references-count":11,"journal-issue":{"issue":"02","published-online":{"date-parts":[[2011,11,20]]},"published-print":{"date-parts":[[2003,4]]}},"alternative-id":["10.1142\/S0218196703001316"],"URL":"https:\/\/doi.org\/10.1142\/s0218196703001316","relation":{},"ISSN":["0218-1967","1793-6500"],"issn-type":[{"value":"0218-1967","type":"print"},{"value":"1793-6500","type":"electronic"}],"subject":[],"published":{"date-parts":[[2003,4]]}}}