{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,8]],"date-time":"2025-10-08T15:23:36Z","timestamp":1759937016586},"reference-count":15,"publisher":"Walter de Gruyter GmbH","issue":"5","license":[{"start":{"date-parts":[[2021,3,11]],"date-time":"2021-03-11T00:00:00Z","timestamp":1615420800000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100001807","name":"Funda\u00e7\u00e3o de Amparo \u00e0 Pesquisa do Estado de S\u00e3o Paulo","doi-asserted-by":"publisher","award":["2016\/12196-5"],"award-info":[{"award-number":["2016\/12196-5"]}],"id":[{"id":"10.13039\/501100001807","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003593","name":"Conselho Nacional de Desenvolvimento Cient\u00edfico e Tecnol\u00f3gico","doi-asserted-by":"publisher","award":["306614\/2016-2"],"award-info":[{"award-number":["306614\/2016-2"]}],"id":[{"id":"10.13039\/501100003593","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","doi-asserted-by":"publisher","award":["UIDB\/04621\/2020","UIDP\/04621\/2020","UIDB\/00144\/2020"],"award-info":[{"award-number":["UIDB\/04621\/2020","UIDP\/04621\/2020","UIDB\/00144\/2020"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2021,9,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>In this work, we study automorphisms of synchronous self-similar groups and the existence of extensions to continuous automorphisms over the closure of these groups with respect to the depth metric.\nWe obtain conditions for the continuity of such extensions, but we also construct examples of groups where such extensions do not exist.\nWe study in detail the case of the lamplighter group <jats:inline-formula>\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:msub>\n                                 <m:mi mathvariant=\"script\">L<\/m:mi>\n                                 <m:mi>k<\/m:mi>\n                              <\/m:msub>\n                              <m:mo>=<\/m:mo>\n                              <m:mrow>\n                                 <m:msub>\n                                    <m:mi mathvariant=\"double-struck\">Z<\/m:mi>\n                                    <m:mi>k<\/m:mi>\n                                 <\/m:msub>\n                                 <m:mo>\u2240<\/m:mo>\n                                 <m:mi mathvariant=\"double-struck\">Z<\/m:mi>\n                              <\/m:mrow>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_jgth-2020-0034_ineq_0001.png\" \/>\n                        <jats:tex-math>\\mathcal{L}_{k}=\\mathbb{Z}_{k}\\wr\\mathbb{Z}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>.<\/jats:p>","DOI":"10.1515\/jgth-2020-0034","type":"journal-article","created":{"date-parts":[[2021,3,11]],"date-time":"2021-03-11T22:51:53Z","timestamp":1615503113000},"page":"857-897","source":"Crossref","is-referenced-by-count":6,"title":["Extensions of automorphisms of self-similar groups"],"prefix":"10.1515","volume":"24","author":[{"given":"Francesco","family":"Matucci","sequence":"first","affiliation":[{"name":"Dipartimento di Matematica e Applicazioni , Universit\u00e0 degli Studi di Milano-Bicocca , Office 3043, Via Cozzi 55, 20125 Milano , Italy"}]},{"given":"Pedro V.","family":"Silva","sequence":"additional","affiliation":[{"name":"Centro de Matem\u00e1tica , Faculdade de Ci\u00eancias , Universidade do Porto , R. Campo Alegre 687, 4169-007 Porto , Portugal"}]}],"member":"374","published-online":{"date-parts":[[2021,3,11]]},"reference":[{"key":"2023040101071097171_j_jgth-2020-0034_ref_001","doi-asserted-by":"crossref","unstructured":"S. Bachmuth, G. Baumslag, J. Dyer and H.\u2009Y. Mochizuki,\nAutomorphism groups of two generator metabelian groups,\nJ. Lond. Math. Soc. (2) 36 (1987), no. 3, 393\u2013406.","DOI":"10.1112\/jlms\/s2-36.3.393"},{"key":"2023040101071097171_j_jgth-2020-0034_ref_002","doi-asserted-by":"crossref","unstructured":"L. Bartholdi, R. Grigorchuk and V. Nekrashevych,\nFrom fractal groups to fractal sets,\nFractals in Graz 2001,\nTrends Math.,\nBirkh\u00e4user, Basel (2003), 25\u2013118.","DOI":"10.1007\/978-3-0348-8014-5_2"},{"key":"2023040101071097171_j_jgth-2020-0034_ref_003","doi-asserted-by":"crossref","unstructured":"L. Bartholdi, R.\u2009I. Grigorchuk and Z. \u0160uni\u1e31,\nBranch groups,\nHandbook of Algebra. Vol. 3,\nElsevier\/North-Holland, Amsterdam (2003), 989\u20131112.","DOI":"10.1016\/S1570-7954(03)80078-5"},{"key":"2023040101071097171_j_jgth-2020-0034_ref_004","doi-asserted-by":"crossref","unstructured":"L. Bartholdi and S.\u2009N. Sidki,\nThe automorphism tower of groups acting on rooted trees,\nTrans. Amer. Math. Soc. 358 (2006), no. 1, 329\u2013358.","DOI":"10.1090\/S0002-9947-05-03712-8"},{"key":"2023040101071097171_j_jgth-2020-0034_ref_005","unstructured":"L. Bartholdi and P. Silva,\nGroups defined by automata, preprint (2010), https:\/\/arxiv.org\/abs\/1012.1531v1; to appear in Handbook of Automata Theory, Chapter 24."},{"key":"2023040101071097171_j_jgth-2020-0034_ref_006","doi-asserted-by":"crossref","unstructured":"L. Bartholdi and Z. \u0160uni\u1e31,\nSome solvable automaton groups,\nTopological and Asymptotic Aspects of Group Theory,\nContemp. Math. 394,\nAmerican Mathematical Society, Providence (2006), 11\u201329.","DOI":"10.1090\/conm\/394\/07431"},{"key":"2023040101071097171_j_jgth-2020-0034_ref_007","doi-asserted-by":"crossref","unstructured":"A.\u2009M. Brunner and S. Sidki,\nOn the automorphism group of the one-rooted binary tree,\nJ. Algebra 195 (1997), no. 2, 465\u2013486.","DOI":"10.1006\/jabr.1997.7033"},{"key":"2023040101071097171_j_jgth-2020-0034_ref_008","doi-asserted-by":"crossref","unstructured":"A.\u2009M. Brunner and S. Sidki,\nThe generation of \n                  \n                     \n                        \n                           GL\n                           \u2062\n                           \n                              (\n                              n\n                              ,\n                              Z\n                              )\n                           \n                        \n                     \n                     \n                     \\mathrm{GL}(n,\\mathbf{Z})\n                  \n                by finite state automata,\nInternat. J. Algebra Comput. 8 (1998), no. 1, 127\u2013139.","DOI":"10.1142\/S0218196798000077"},{"key":"2023040101071097171_j_jgth-2020-0034_ref_009","doi-asserted-by":"crossref","unstructured":"R.\u2009I. Grigorchuk,\nJust infinite branch groups,\nNew Horizons in Pro-\ud835\udc5d Groups,\nProgr. Math. 184,\nBirkh\u00e4user, Boston (2000), 121\u2013179.","DOI":"10.1007\/978-1-4612-1380-2_4"},{"key":"2023040101071097171_j_jgth-2020-0034_ref_010","unstructured":"K.\u2009B. Krohn and J.\u2009L. Rhodes,\nAlgebraic theory of machines,\nProceedings of the Symposium on Mathematical Theory of Automata (New York 1962),\nPolytechnic Press, Brooklyn (1963), 341\u2013384."},{"key":"2023040101071097171_j_jgth-2020-0034_ref_011","doi-asserted-by":"crossref","unstructured":"Y. Lavreniuk and V. Nekrashevych,\nRigidity of branch groups acting on rooted trees,\nGeom. Dedicata 89 (2002), 159\u2013179.","DOI":"10.1023\/A:1014264212978"},{"key":"2023040101071097171_j_jgth-2020-0034_ref_012","doi-asserted-by":"crossref","unstructured":"V. Nekrashevych,\nSelf-Similar Groups,\nMath. Surveys Monogr. 117,\nAmerican Mathematical Society, Providence, 2005.","DOI":"10.1090\/surv\/117\/04"},{"key":"2023040101071097171_j_jgth-2020-0034_ref_013","doi-asserted-by":"crossref","unstructured":"P.\u2009V. Silva and B. Steinberg,\nOn a class of automata groups generalizing lamplighter groups,\nInternat. J. Algebra Comput. 15 (2005), no. 5\u20136, 1213\u20131234.","DOI":"10.1142\/S0218196705002761"},{"key":"2023040101071097171_j_jgth-2020-0034_ref_014","doi-asserted-by":"crossref","unstructured":"M. Stein, J. Taback and P. Wong,\nAutomorphisms of higher rank lamplighter groups,\nInternat. J. Algebra Comput. 25 (2015), no. 8, 1275\u20131299.","DOI":"10.1142\/S0218196715500411"},{"key":"2023040101071097171_j_jgth-2020-0034_ref_015","doi-asserted-by":"crossref","unstructured":"M. Vorobets and Y. Vorobets,\nOn a free group of transformations defined by an automaton,\nGeom. Dedicata 124 (2007), 237\u2013249.","DOI":"10.1007\/s10711-006-9060-5"}],"container-title":["Journal of Group Theory"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/jgth-2020-0034\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/jgth-2020-0034\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,4,1]],"date-time":"2023-04-01T18:11:57Z","timestamp":1680372717000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/jgth-2020-0034\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,3,11]]},"references-count":15,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2021,9,1]]},"published-print":{"date-parts":[[2021,9,1]]}},"alternative-id":["10.1515\/jgth-2020-0034"],"URL":"https:\/\/doi.org\/10.1515\/jgth-2020-0034","relation":{},"ISSN":["1433-5883","1435-4446"],"issn-type":[{"value":"1433-5883","type":"print"},{"value":"1435-4446","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,3,11]]}}}