{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:58:23Z","timestamp":1760061503770},"reference-count":20,"publisher":"World Scientific Pub Co Pte Lt","issue":"07","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Algebra Comput."],"published-print":{"date-parts":[[2008,11]]},"abstract":"<jats:p> Viewing Dehn's algorithm as a rewriting system, we generalize to allow an alphabet containing letters which do not necessarily represent group elements. This extends the class of groups for which the algorithm solves the word problem to include finitely generated nilpotent groups, many relatively hyperbolic groups including geometrically finite groups and fundamental groups of certain geometrically decomposable 3-manifolds. The class has several nice closure properties. We also show that if a group has an infinite subgroup and one of exponential growth, and they commute, then it does not admit such an algorithm. We dub these Cannon's algorithms. <\/jats:p>","DOI":"10.1142\/s0218196708004822","type":"journal-article","created":{"date-parts":[[2008,11,27]],"date-time":"2008-11-27T13:36:08Z","timestamp":1227792968000},"page":"1137-1177","source":"Crossref","is-referenced-by-count":8,"title":["ON A GENERALIZATION OF DEHN'S ALGORITHM"],"prefix":"10.1142","volume":"18","author":[{"given":"OLIVER","family":"GOODMAN","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, University of Melbourne, Parkville, Victoria 3052, Australia"}]},{"given":"MICHAEL","family":"SHAPIRO","sequence":"additional","affiliation":[{"name":"Department of Pathology, Tufts University, Boston, MA 02111, USA"}]}],"member":"219","published-online":{"date-parts":[[2011,11,20]]},"reference":[{"key":"rf1","volume-title":"Group Theory From a Geometric Viewpoint","author":"Alonso J. M.","year":"1991"},{"key":"rf2","doi-asserted-by":"publisher","DOI":"10.1112\/blms\/26.1.97"},{"key":"rf4","doi-asserted-by":"publisher","DOI":"10.1006\/inco.1997.2681"},{"key":"rf5","first-page":"123","volume":"16","author":"Cannon J.","journal-title":"Geom. Dedicata"},{"key":"rf6","unstructured":"J.\u00a0Cannon, Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces, eds. T.\u00a0Bedford, M.\u00a0Keane and C.\u00a0Series (Oxford University Press, 1991)\u00a0pp. 315\u2013369."},{"key":"rf8","first-page":"215","volume":"42","author":"Cannon J. W.","journal-title":"Enseign. Math."},{"key":"rf9","doi-asserted-by":"publisher","DOI":"10.2140\/gt.2003.7.933"},{"key":"rf10","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-4668-8"},{"key":"rf11","doi-asserted-by":"publisher","DOI":"10.1016\/0196-6774(85)90031-8"},{"key":"rf13","first-page":"810","volume":"8","author":"Farb B.","journal-title":"GAFA"},{"key":"rf14","doi-asserted-by":"publisher","DOI":"10.1007\/BF02698687"},{"key":"rf15","doi-asserted-by":"publisher","DOI":"10.1142\/S0218196700000078"},{"key":"rf16","doi-asserted-by":"publisher","DOI":"10.1017\/S0024610701002083"},{"key":"rf18","volume-title":"Introduction to Automata Theory, Languages and Computation","author":"Hopcroft J.","year":"1979"},{"key":"rf21","volume":"53","author":"Lysionok I. G.","journal-title":"Izv. Akad. Nauk Ser. Math."},{"key":"rf22","doi-asserted-by":"publisher","DOI":"10.1007\/BF01241129"},{"key":"rf23","volume-title":"Relatively hyperbolic groups: Intrinsic geometry, algebraic properties, and algorithmic problems","author":"Osin D. V.","year":"2006"},{"key":"rf24","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511565953"},{"key":"rf25","doi-asserted-by":"publisher","DOI":"10.1007\/s10240-001-8188-y"},{"key":"rf26","doi-asserted-by":"publisher","DOI":"10.1142\/S0218196794000129"}],"container-title":["International Journal of Algebra and Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218196708004822","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,6]],"date-time":"2019-08-06T22:20:09Z","timestamp":1565130009000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0218196708004822"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,11]]},"references-count":20,"journal-issue":{"issue":"07","published-online":{"date-parts":[[2011,11,20]]},"published-print":{"date-parts":[[2008,11]]}},"alternative-id":["10.1142\/S0218196708004822"],"URL":"https:\/\/doi.org\/10.1142\/s0218196708004822","relation":{},"ISSN":["0218-1967","1793-6500"],"issn-type":[{"value":"0218-1967","type":"print"},{"value":"1793-6500","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,11]]}}}