{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,14]],"date-time":"2026-01-14T23:47:44Z","timestamp":1768434464524,"version":"3.49.0"},"reference-count":21,"publisher":"World Scientific Pub Co Pte Ltd","issue":"08","funder":[{"name":"Russian Foundation for Basic Research","award":["20-01-00465"],"award-info":[{"award-number":["20-01-00465"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Algebra Comput."],"published-print":{"date-parts":[[2022,12]]},"abstract":"<jats:p> In this paper, we introduce a concept of an evacuation scheme on the Cayley graph of an infinite finitely generated group. This is a collection of infinite simple paths bringing all vertices to infinity. We impose a restriction that every edge can be used a uniformly bounded number of times in this scheme. An easy observation shows that the existence of such a scheme is equivalent to non-amenability of the group. A special case happens if every edge can be used only once. These schemes are called pure. We obtain a criterion for the existence of such a scheme in terms of isoperimetric constant of the graph. We analyze R. Thompson\u2019s group [Formula: see text], for which the amenability property is a famous open problem. We show that pure evacuation schemes do not exist for the set of generators [Formula: see text], where [Formula: see text]. However, the question becomes open if edges with labels [Formula: see text] can be used twice. The existence of pure evacuation schemes for this version is implied by some natural conjectures. <\/jats:p>","DOI":"10.1142\/s0218196722500667","type":"journal-article","created":{"date-parts":[[2022,9,9]],"date-time":"2022-09-09T15:52:33Z","timestamp":1662738753000},"page":"1477-1494","source":"Crossref","is-referenced-by-count":1,"title":["Evacuation schemes on Cayley graphs and non-amenability of groups"],"prefix":"10.1142","volume":"32","author":[{"given":"Victor","family":"Guba","sequence":"first","affiliation":[{"name":"Department of Applied Mathematics, Vologda State University, 15 Lenin Street, Vologda 160600, Russia"}]}],"member":"219","published-online":{"date-parts":[[2022,10,15]]},"reference":[{"issue":"5","key":"S0218196722500667BIB003","doi-asserted-by":"crossref","first-page":"815","DOI":"10.1142\/S021819670500261X","volume":"15","author":"Belk J. M.","year":"2005","journal-title":"Internat. J. Algebra Comput."},{"key":"S0218196722500667BIB004","doi-asserted-by":"publisher","DOI":"10.1007\/BF01388519"},{"key":"S0218196722500667BIB005","doi-asserted-by":"publisher","DOI":"10.1016\/0022-4049(87)90015-6"},{"key":"S0218196722500667BIB006","doi-asserted-by":"publisher","DOI":"10.1007\/BF01388451"},{"issue":"2","key":"S0218196722500667BIB007","first-page":"215","volume":"42","author":"Cannon J. W.","year":"1996","journal-title":"Enseign. Math."},{"key":"S0218196722500667BIB009","doi-asserted-by":"publisher","DOI":"10.1215\/ijm\/1255380675"},{"key":"S0218196722500667BIB010","first-page":"17","volume":"30","author":"Deuber W. A.","year":"1995","journal-title":"Studia Math."},{"key":"S0218196722500667BIB011","doi-asserted-by":"publisher","DOI":"10.7146\/math.scand.a-10442"},{"key":"S0218196722500667BIB012","doi-asserted-by":"publisher","DOI":"10.4153\/CJM-1956-045-5"},{"key":"S0218196722500667BIB013","series-title":"Annals of Mathematics Studies","doi-asserted-by":"crossref","first-page":"545","DOI":"10.1515\/9781400882083-026","volume-title":"Combinatorial Group Theory and Topology","volume":"111","author":"Gersten S. M.","year":"1987"},{"key":"S0218196722500667BIB014","volume-title":"Invariant Means on Topological Groups and Their Applications","author":"Greenleaf F. P.","year":"1969"},{"issue":"5","key":"S0218196722500667BIB016","doi-asserted-by":"crossref","first-page":"677","DOI":"10.1142\/S021819670400192X","volume":"14","author":"Guba V.","year":"2004","journal-title":"Internat. J. Algebra Comput."},{"issue":"5","key":"S0218196722500667BIB017","doi-asserted-by":"crossref","first-page":"969","DOI":"10.1142\/S0218196721500454","volume":"31","author":"Guba V.","year":"2021","journal-title":"Internat. J. Algebra Comput."},{"key":"S0218196722500667BIB018","doi-asserted-by":"crossref","first-page":"315","DOI":"10.1017\/S1446788700001038","volume":"62","author":"Guba V. S.","year":"1997","journal-title":"J. Austral. Math. Soc. (Ser. A)"},{"key":"S0218196722500667BIB019","doi-asserted-by":"publisher","DOI":"10.1090\/memo\/0620"},{"key":"S0218196722500667BIB021","first-page":"121","volume":"3","author":"K\u00f6nig D.","year":"1927","journal-title":"Acta Sci. Math. (Szeged)"},{"key":"S0218196722500667BIB022","doi-asserted-by":"publisher","DOI":"10.4171\/GGD\/201"},{"key":"S0218196722500667BIB023","doi-asserted-by":"publisher","DOI":"10.4064\/fm-13-1-73-116"},{"key":"S0218196722500667BIB025","doi-asserted-by":"publisher","DOI":"10.1007\/s10240-002-0006-7"},{"key":"S0218196722500667BIB026","series-title":"Springer Monographs in Mathematics","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-319-08031-4","volume-title":"Combinatorial Algebra: Syntax and Semantics.","author":"Sapir M.","year":"2014"},{"key":"S0218196722500667BIB027","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-61856-7"}],"container-title":["International Journal of Algebra and Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218196722500667","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,12,27]],"date-time":"2022-12-27T03:23:00Z","timestamp":1672111380000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/10.1142\/S0218196722500667"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,10,15]]},"references-count":21,"journal-issue":{"issue":"08","published-print":{"date-parts":[[2022,12]]}},"alternative-id":["10.1142\/S0218196722500667"],"URL":"https:\/\/doi.org\/10.1142\/s0218196722500667","relation":{},"ISSN":["0218-1967","1793-6500"],"issn-type":[{"value":"0218-1967","type":"print"},{"value":"1793-6500","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,10,15]]}}}