{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,10]],"date-time":"2026-04-10T16:57:15Z","timestamp":1775840235455,"version":"3.50.1"},"reference-count":42,"publisher":"American Mathematical Society (AMS)","issue":"2","license":[{"start":{"date-parts":[[2013,9,24]],"date-time":"2013-09-24T00:00:00Z","timestamp":1379980800000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Trans. Amer. Math. Soc."],"abstract":"<p>We study groups acting by length-preserving transformations on spaces equipped with asymmetric, partially-defined distance functions. We introduce a natural notion of quasi-isometry for such spaces and exhibit an extension of the \u0160varc-Milnor lemma to this setting. Among the most natural examples of these spaces are finitely generated monoids and semigroups and their Cayley and Sch\u00fctzenberger graphs. We apply our results to show that a number of important properties of monoids are quasi-isometry invariants.<\/p>","DOI":"10.1090\/s0002-9947-2012-05868-5","type":"journal-article","created":{"date-parts":[[2012,9,24]],"date-time":"2012-09-24T10:43:56Z","timestamp":1348483436000},"page":"555-578","source":"Crossref","is-referenced-by-count":9,"special_numbering":"933","title":["Groups acting on semimetric spaces and quasi-isometries of monoids"],"prefix":"10.1090","volume":"365","author":[{"given":"Robert","family":"Gray","sequence":"first","affiliation":[]},{"given":"Mark","family":"Kambites","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2012,9,24]]},"reference":[{"key":"1","volume-title":"Algebraic theory of machines, languages, and semigroups","year":"1968"},{"issue":"12","key":"2","first-page":"761","article-title":"In\u00e9galit\u00e9s isop\u00e9rim\u00e9triques et quasi-isom\u00e9tries","volume":"311","author":"Alonso, Juan M.","year":"1990","journal-title":"C. 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