{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,5,8]],"date-time":"2023-05-08T06:29:11Z","timestamp":1683527351063},"reference-count":5,"publisher":"World Scientific Pub Co Pte Lt","issue":"01n02","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Algebra Comput."],"published-print":{"date-parts":[[2011,2]]},"abstract":"<jats:p> We introduce the peak normal form for elements of the Baumslag\u2013Solitar groups BS(p,q). This normal form is very close to the length-lexicographical normal form, but more symmetric. Both normal forms are geodesic. This means the normal form of an element u<jats:sup>-1<\/jats:sup>v yields the shortest path between u and v in the Cayley graph. For horocyclic elements the peak normal form and the length-lexicographical normal form coincide. The main result of this paper is that we can compute the peak normal form in polynomial time if p divides q. As consequence we can compute geodesic lengths in this case. In particular, this gives a partial answer to Question 1 in [4] <\/jats:p><jats:p> For arbitrary p and q it is possible to compute the peak normal form (length-lexicographical normal form resp.) also the for elements in the horocyclic subgroup and, more generally, for elements which we call hills. This approach leads to a linear time reduction of the problem of computing geodesics to the problem of computing geodesics for Britton-reduced words where the t-sequence starts with t<jats:sup>-1<\/jats:sup> and ends with t. <\/jats:p>","DOI":"10.1142\/s0218196711006108","type":"journal-article","created":{"date-parts":[[2011,4,6]],"date-time":"2011-04-06T08:14:25Z","timestamp":1302077665000},"page":"119-145","source":"Crossref","is-referenced-by-count":4,"title":["ON COMPUTING GEODESICS IN BAUMSLAG\u2013SOLITAR GROUPS"],"prefix":"10.1142","volume":"21","author":[{"given":"VOLKER","family":"DIEKERT","sequence":"first","affiliation":[{"name":"Universit\u00e4t Stuttgart, FMI, Universit\u00e4tsstra\u00dfe 38, D-70569 Stuttgart, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"J\u00dcRN","family":"LAUN","sequence":"additional","affiliation":[{"name":"Universit\u00e4t Stuttgart, FMI, Universit\u00e4tsstra\u00dfe 38, D-70569 Stuttgart, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2011,11,20]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9904-1962-10745-9"},{"key":"rf2","series-title":"Trends in Mathematics","volume-title":"Combinatorial and Geometric Group Theory","author":"Diekert V.","year":"2009"},{"key":"rf7","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-61549-8"},{"key":"rf8","doi-asserted-by":"publisher","DOI":"10.1016\/S0019-9958(70)90105-1"},{"key":"rf9","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-61896-3"}],"container-title":["International Journal of Algebra and Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218196711006108","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,6]],"date-time":"2019-08-06T18:21:53Z","timestamp":1565115713000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0218196711006108"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,2]]},"references-count":5,"journal-issue":{"issue":"01n02","published-online":{"date-parts":[[2011,11,20]]},"published-print":{"date-parts":[[2011,2]]}},"alternative-id":["10.1142\/S0218196711006108"],"URL":"https:\/\/doi.org\/10.1142\/s0218196711006108","relation":{},"ISSN":["0218-1967","1793-6500"],"issn-type":[{"value":"0218-1967","type":"print"},{"value":"1793-6500","type":"electronic"}],"subject":[],"published":{"date-parts":[[2011,2]]}}}