{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,5]],"date-time":"2025-10-05T04:35:55Z","timestamp":1759638955452,"version":"3.37.3"},"reference-count":24,"publisher":"World Scientific Pub Co Pte Ltd","issue":"03","funder":[{"DOI":"10.13039\/501100000923","name":"Australian Research Council","doi-asserted-by":"crossref","award":["DP160100486"],"award-info":[{"award-number":["DP160100486"]}],"id":[{"id":"10.13039\/501100000923","id-type":"DOI","asserted-by":"crossref"}]},{"DOI":"10.13039\/100000893","name":"Simons Foundation","doi-asserted-by":"crossref","award":["31736"],"award-info":[{"award-number":["31736"]}],"id":[{"id":"10.13039\/100000893","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Algebra Comput."],"published-print":{"date-parts":[[2022,5]]},"abstract":"<jats:p> In contrast to being automatic, being Cayley automatic a priori has no geometric consequences. Specifically, Cayley graphs of automatic groups enjoy a fellow traveler property. Here, we study a distance function introduced by the first author and Trakuldit which aims to measure how far a Cayley automatic group is from being automatic, in terms of how badly the Cayley graph fails the fellow traveler property. The first author and Trakuldit showed that if it fails by at most a constant amount, then the group is in fact automatic. In this paper, we show that for a large class of non-automatic Cayley automatic groups this function is bounded below by a linear function in a precise sense defined herein. In fact, for all Cayley automatic groups which have super-quadratic Dehn function, or which are not finitely presented, we can construct a non-decreasing function which (1) depends only on the group and (2) bounds from below the distance function for any Cayley automatic structure on the group. <\/jats:p>","DOI":"10.1142\/s0218196722500199","type":"journal-article","created":{"date-parts":[[2022,3,16]],"date-time":"2022-03-16T10:33:42Z","timestamp":1647426822000},"page":"383-409","source":"Crossref","is-referenced-by-count":2,"title":["On the geometry of Cayley automatic groups"],"prefix":"10.1142","volume":"32","author":[{"given":"Dmitry","family":"Berdinsky","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Mahidol University, Bangkok 10400, Thailand"},{"name":"Centre of Excellence in Mathematics, Commission on Higher Education, Bangkok 10400, Thailand"}]},{"given":"Murray","family":"Elder","sequence":"additional","affiliation":[{"name":"School of Mathematical and Physical Sciences, University of Technology Sydney, Ultimo, NSW 2007, Australia"}]},{"given":"Jennifer","family":"Taback","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Bowdoin College, 8600 College Station, Brunswick, ME 04011, USA"}]}],"member":"219","published-online":{"date-parts":[[2022,3,15]]},"reference":[{"key":"S0218196722500199BIB001","doi-asserted-by":"publisher","DOI":"10.1007\/s000390050053"},{"key":"S0218196722500199BIB002","doi-asserted-by":"publisher","DOI":"10.4171\/JEMS\/130"},{"key":"S0218196722500199BIB003","doi-asserted-by":"publisher","DOI":"10.1007\/BF01211007"},{"key":"S0218196722500199BIB004","doi-asserted-by":"publisher","DOI":"10.1017\/S000497272100023X"},{"key":"S0218196722500199BIB005","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-09698-8_1"},{"key":"S0218196722500199BIB006","doi-asserted-by":"publisher","DOI":"10.1142\/S0129054116400049"},{"key":"S0218196722500199BIB007","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-77313-1_19"},{"key":"S0218196722500199BIB008","doi-asserted-by":"publisher","DOI":"10.1016\/j.jalgebra.2017.09.038"},{"key":"S0218196722500199BIB009","doi-asserted-by":"publisher","DOI":"10.1007\/s00224-004-1133-y"},{"key":"S0218196722500199BIB010","doi-asserted-by":"publisher","DOI":"10.1307\/mmj\/1029005156"},{"key":"S0218196722500199BIB011","doi-asserted-by":"publisher","DOI":"10.1007\/s00208-016-1470-6"},{"key":"S0218196722500199BIB012","doi-asserted-by":"publisher","DOI":"10.1007\/s00039-009-0011-9"},{"key":"S0218196722500199BIB013","doi-asserted-by":"publisher","DOI":"10.1016\/j.jalgebra.2014.04.021"},{"key":"S0218196722500199BIB014","doi-asserted-by":"publisher","DOI":"10.1201\/9781439865699"},{"key":"S0218196722500199BIB016","series-title":"CRM Proceedings and Lecture Notes","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1090\/crmp\/017\/03","volume-title":"Summer School in Group Theory in Banff, 1996","volume":"17","author":"Gersten S. 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