{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,4,2]],"date-time":"2022-04-02T11:10:30Z","timestamp":1648897830804},"reference-count":26,"publisher":"World Scientific Pub Co Pte Lt","issue":"05","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Algebra Comput."],"published-print":{"date-parts":[[2011,8]]},"abstract":"<jats:p> A conjugation-free geometric presentation of a fundamental group is a presentation with the natural topological generators x<jats:sub>1<\/jats:sub>,\u2026,x<jats:sub>n<\/jats:sub> and the cyclic relations: [Formula: see text] with no conjugations on the generators. <\/jats:p><jats:p> We have already proved in [13] that if the graph of the arrangement is a disjoint union of cycles, then its fundamental group has a conjugation-free geometric presentation. In this paper, we extend this property to arrangements whose graphs are a disjoint union of cycle-tree graphs. <\/jats:p><jats:p> Moreover, we study some properties of this type of presentations for a fundamental group of a line arrangement's complement. We show that these presentations satisfy a completeness property in the sense of Dehornoy, if the corresponding graph of the arrangement is triangle-free. The completeness property is a powerful property which leads to many nice properties concerning the presentation (such as the left-cancellativity of the associated monoid and yields some simple criterion for the solvability of the word problem in the group). <\/jats:p>","DOI":"10.1142\/s0218196711006479","type":"journal-article","created":{"date-parts":[[2011,3,28]],"date-time":"2011-03-28T11:33:30Z","timestamp":1301312010000},"page":"775-792","source":"Crossref","is-referenced-by-count":5,"title":["A CONJUGATION-FREE GEOMETRIC PRESENTATION OF FUNDAMENTAL GROUPS OF ARRANGEMENTS II: EXPANSION AND SOME PROPERTIES"],"prefix":"10.1142","volume":"21","author":[{"given":"MEITAL","family":"ELIYAHU","sequence":"first","affiliation":[{"name":"Department of Mathematics, Bar-Ilan University, 52900 Ramat-Gan, Israel"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"DAVID","family":"GARBER","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, Faculty of Sciences, Holon Institute of Technology, 52 Golomb st., PO Box 305, 58102 Holon, Israel"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"MINA","family":"TEICHER","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Bar-Ilan University, 52900 Ramat-Gan, Israel"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2011,11,20]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.1007\/BF02677471"},{"key":"rf2","first-page":"13","volume":"83","author":"Artal-Bartolo E.","journal-title":"Topol. 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