{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,11,15]],"date-time":"2022-11-15T06:09:04Z","timestamp":1668492544488},"reference-count":27,"publisher":"Walter de Gruyter GmbH","issue":"0","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>We describe the endomorphisms of the direct product of two free groups of finite rank and obtain conditions for which the subgroup of fixed points is finitely generated, and we do the same for periodic points.\nWe also describe the automorphisms of such a group and solve the three Whitehead problems for automorphisms, monomorphisms and endomorphisms for groups in this class.\nWe also obtain conditions for an endomorphism to be uniformly continuous when a suitable metric is considered and study infinite fixed points of the extension to the completion of the group.<\/jats:p>","DOI":"10.1515\/jgth-2021-0197","type":"journal-article","created":{"date-parts":[[2022,11,14]],"date-time":"2022-11-14T22:01:08Z","timestamp":1668463268000},"source":"Crossref","is-referenced-by-count":0,"title":["On endomorphisms of the direct product  of two free groups"],"prefix":"10.1515","volume":"0","author":[{"given":"Andr\u00e9","family":"Carvalho","sequence":"first","affiliation":[{"name":"Centre of Mathematics , University of Porto , R. Campo Alegre, 4169-007 Porto , Portugal"}]}],"member":"374","published-online":{"date-parts":[[2022,11,15]]},"reference":[{"key":"2022111422010303809_j_jgth-2021-0197_ref_001","doi-asserted-by":"crossref","unstructured":"J. Berstel,\nTransductions and Context-Free Languages,\nGuides Appl. Math. Mech. 38,\nB.\u2009G., Teubner, Stuttgart, 1979.","DOI":"10.1007\/978-3-663-09367-1"},{"key":"2022111422010303809_j_jgth-2021-0197_ref_002","doi-asserted-by":"crossref","unstructured":"M. Bestvina and M. Handel,\nTrain tracks and automorphisms of free groups,\nAnn. of Math. (2) 135 (1992), no. 1, 1\u201351.","DOI":"10.2307\/2946562"},{"key":"2022111422010303809_j_jgth-2021-0197_ref_003","doi-asserted-by":"crossref","unstructured":"O. Bogopolski and O. Maslakova,\nAn algorithm for finding a basis of the fixed point subgroup of an automorphism of a free group,\nInternat. J. Algebra Comput. 26 (2016), no. 1, 29\u201367.","DOI":"10.1142\/S0218196716500028"},{"key":"2022111422010303809_j_jgth-2021-0197_ref_004","doi-asserted-by":"crossref","unstructured":"A. Carvalho,\nOn the dynamics of extensions of free-abelian times free groups endomorphisms to the completion,\nProc. Edinburgh Math. Soc. 65 (2022), no. 3, 705\u2013735.","DOI":"10.1017\/S001309152200030X"},{"key":"2022111422010303809_j_jgth-2021-0197_ref_005","unstructured":"A. Carvalho,\nEventually fixed points of endomorphisms of virtually free groups,\npreprint (2022), https:\/\/arxiv.org\/abs\/2204.04543."},{"key":"2022111422010303809_j_jgth-2021-0197_ref_006","doi-asserted-by":"crossref","unstructured":"J. Cassaigne and P.\u2009V. Silva,\nInfinite words and confluent rewriting systems: Endomorphism extensions,\nInternat. J. Algebra Comput. 19 (2009), no. 4, 443\u2013490.","DOI":"10.1142\/S0218196709005111"},{"key":"2022111422010303809_j_jgth-2021-0197_ref_007","doi-asserted-by":"crossref","unstructured":"L. Ciobanu and A. Ould Houcine,\nThe monomorphism problem in free groups,\nArch. Math. (Basel) 94 (2010), no. 5, 423\u2013434.","DOI":"10.1007\/s00013-010-0118-0"},{"key":"2022111422010303809_j_jgth-2021-0197_ref_008","doi-asserted-by":"crossref","unstructured":"D. Cooper,\nAutomorphisms of free groups have finitely generated fixed point sets,\nJ. Algebra 111 (1987), no. 2, 453\u2013456.","DOI":"10.1016\/0021-8693(87)90229-8"},{"key":"2022111422010303809_j_jgth-2021-0197_ref_009","doi-asserted-by":"crossref","unstructured":"M.\u2009B. Day,\nPeak reduction and finite presentations for automorphism groups of right-angled Artin groups,\nGeom. Topol. 13 (2009), no. 2, 817\u2013855.","DOI":"10.2140\/gt.2009.13.817"},{"key":"2022111422010303809_j_jgth-2021-0197_ref_010","doi-asserted-by":"crossref","unstructured":"M.\u2009B. Day,\nFull-featured peak reduction in right-angled Artin groups,\nAlgebr. Geom. Topol. 14 (2014), no. 3, 1677\u20131743.","DOI":"10.2140\/agt.2014.14.1677"},{"key":"2022111422010303809_j_jgth-2021-0197_ref_011","doi-asserted-by":"crossref","unstructured":"J. Delgado and E. Ventura,\nAlgorithmic problems for free-abelian times free groups,\nJ. Algebra 391 (2013), 256\u2013283.","DOI":"10.1016\/j.jalgebra.2013.04.033"},{"key":"2022111422010303809_j_jgth-2021-0197_ref_012","unstructured":"V. Diekert, A. Je\u017c and M. Kufleitner,\nSolutions of word equations over partially commutative structures,\n43rd International Colloquium on Automata, Languages, and Programming,\nLIPIcs. Leibniz Int. Proc. Inform. 55,\nSchloss Dagstuhl, Wadern (2016), Article ID 127."},{"key":"2022111422010303809_j_jgth-2021-0197_ref_013","doi-asserted-by":"crossref","unstructured":"D. Gaboriau, A. Jaeger, G. Levitt and M. Lustig,\nAn index for counting fixed points of automorphisms of free groups,\nDuke Math. J. 93 (1998), no. 3, 425\u2013452.","DOI":"10.1215\/S0012-7094-98-09314-0"},{"key":"2022111422010303809_j_jgth-2021-0197_ref_014","doi-asserted-by":"crossref","unstructured":"S.\u2009M. Gersten,\nFixed points of automorphisms of free groups,\nAdv. in Math. 64 (1987), no. 1, 51\u201385.","DOI":"10.1016\/0001-8708(87)90004-1"},{"key":"2022111422010303809_j_jgth-2021-0197_ref_015","doi-asserted-by":"crossref","unstructured":"A.\u2009G. Howson,\nOn the intersection of finitely generated free groups,\nJ. Lond. Math. Soc. 29 (1954), 428\u2013434.","DOI":"10.1112\/jlms\/s1-29.4.428"},{"key":"2022111422010303809_j_jgth-2021-0197_ref_016","doi-asserted-by":"crossref","unstructured":"G. Levitt and M. Lustig,\nAutomorphisms of free groups have asymptotically periodic dynamics,\nJ. Reine Angew. Math. 619 (2008), 1\u201336.","DOI":"10.1515\/CRELLE.2008.038"},{"key":"2022111422010303809_j_jgth-2021-0197_ref_017","unstructured":"M. Lothaire,\nCombinatorics on Words,\nEncyclopedia Math. Appl. 17,\nAddison-Wesley, Reading, 1983."},{"key":"2022111422010303809_j_jgth-2021-0197_ref_018","unstructured":"G.\u2009S. Makanin,\nEquations in a free group,\nIzv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 6, 1199\u20131273, 1344."},{"key":"2022111422010303809_j_jgth-2021-0197_ref_019","doi-asserted-by":"crossref","unstructured":"F. Matucci and P.\u2009V. Silva,\nExtensions of automorphisms of self-similar groups,\nJ. Group Theory 24 (2021), no. 5, 857\u2013897.","DOI":"10.1515\/jgth-2020-0034"},{"key":"2022111422010303809_j_jgth-2021-0197_ref_020","unstructured":"K.\u2009A. Miha\u012dlova,\nThe occurrence problem for direct products of groups,\nDokl. Akad. Nauk SSSR 119 (1958), 1103\u20131105."},{"key":"2022111422010303809_j_jgth-2021-0197_ref_021","unstructured":"J.\u2009P. Mutanguha,\nConstructing stable images,\npreprint (2022), https:\/\/mutanguha.com\/pdfs\/relimmalgo.pdf."},{"key":"2022111422010303809_j_jgth-2021-0197_ref_022","doi-asserted-by":"crossref","unstructured":"A.\u2009G. Myasnikov and V. Shpilrain,\nAutomorphic orbits in free groups,\nJ. Algebra 269 (2003), no. 1, 18\u201327.","DOI":"10.1016\/S0021-8693(03)00339-9"},{"key":"2022111422010303809_j_jgth-2021-0197_ref_023","doi-asserted-by":"crossref","unstructured":"F. Paulin,\nPoints fixes des automorphismes de groupe hyperbolique,\nAnn. Inst. Fourier (Grenoble) 39 (1989), no. 3, 651\u2013662.","DOI":"10.5802\/aif.1181"},{"key":"2022111422010303809_j_jgth-2021-0197_ref_024","doi-asserted-by":"crossref","unstructured":"E. Rodaro, P.\u2009V. Silva and M. Sykiotis,\nFixed points of endomorphisms of graph groups,\nJ. Group Theory 16 (2013), no. 4, 573\u2013583.","DOI":"10.1515\/jgt-2012-0047"},{"key":"2022111422010303809_j_jgth-2021-0197_ref_025","doi-asserted-by":"crossref","unstructured":"P.\u2009V. Silva,\nFixed points of endomorphisms of virtually free groups,\nPacific J. Math. 263 (2013), no. 1, 207\u2013240.","DOI":"10.2140\/pjm.2013.263.207"},{"key":"2022111422010303809_j_jgth-2021-0197_ref_026","doi-asserted-by":"crossref","unstructured":"J.\u2009H.\u2009C. Whitehead,\nOn equivalent sets of elements in a free group,\nAnn. of Math. (2) 37 (1936), no. 4, 782\u2013800.","DOI":"10.2307\/1968618"},{"key":"2022111422010303809_j_jgth-2021-0197_ref_027","unstructured":"S. Willard,\nGeneral Topology,\nAddison-Wesley, Reading, 1970."}],"container-title":["Journal of Group Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/jgth-2021-0197\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/jgth-2021-0197\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,11,14]],"date-time":"2022-11-14T22:01:15Z","timestamp":1668463275000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/jgth-2021-0197\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,11,15]]},"references-count":27,"journal-issue":{"issue":"0","published-online":{"date-parts":[[2022,11,15]]}},"alternative-id":["10.1515\/jgth-2021-0197"],"URL":"https:\/\/doi.org\/10.1515\/jgth-2021-0197","relation":{},"ISSN":["1433-5883","1435-4446"],"issn-type":[{"value":"1433-5883","type":"print"},{"value":"1435-4446","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,11,15]]}}}