{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,12,20]],"date-time":"2023-12-20T16:32:26Z","timestamp":1703089946059},"reference-count":15,"publisher":"World Scientific Pub Co Pte Lt","issue":"05","funder":[{"name":"Russian Foundation for Basic Research","award":["19-01-00591 A"],"award-info":[{"award-number":["19-01-00591 A"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Algebra Comput."],"published-print":{"date-parts":[[2021,8]]},"abstract":"<jats:p> By the density of a finite graph we mean its average vertex degree. For an [Formula: see text]-generated group, the density of its Cayley graph in a given set of generators, is the supremum of densities taken over all its finite subgraphs. It is known that a group with [Formula: see text] generators is amenable if and only if the density of the corresponding Cayley graph equals [Formula: see text]. <\/jats:p><jats:p> A famous problem on the amenability of R. Thompson\u2019s group [Formula: see text] is still open. Due to the result of Belk and Brown, it is known that the density of its Cayley graph in the standard set of group generators [Formula: see text], is at least [Formula: see text]. This estimate has not been exceeded so far. <\/jats:p><jats:p> For the set of symmetric generators [Formula: see text], where [Formula: see text], the same example only gave an estimate of [Formula: see text]. There was a conjecture that for this generating set equality holds. If so, [Formula: see text] would be non-amenable, and the symmetric generating set would have the doubling property. This would mean that for any finite set [Formula: see text], the inequality [Formula: see text] holds. <\/jats:p><jats:p> In this paper, we disprove this conjecture showing that the density of the Cayley graph of [Formula: see text] in symmetric generators [Formula: see text] strictly exceeds [Formula: see text]. Moreover, we show that even larger generating set [Formula: see text] does not have doubling property. <\/jats:p>","DOI":"10.1142\/s0218196721500454","type":"journal-article","created":{"date-parts":[[2021,5,15]],"date-time":"2021-05-15T16:11:36Z","timestamp":1621095096000},"page":"969-981","source":"Crossref","is-referenced-by-count":3,"title":["On the density of Cayley graphs of R.Thompson\u2019s group F in symmetric generators"],"prefix":"10.1142","volume":"31","author":[{"given":"V. S.","family":"Guba","sequence":"first","affiliation":[{"name":"Vologda State University, 15 Lenin Street, Vologda 160600, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2021,5,12]]},"reference":[{"key":"S0218196721500454BIB002","doi-asserted-by":"publisher","DOI":"10.5802\/ambp.249"},{"key":"S0218196721500454BIB004","doi-asserted-by":"publisher","DOI":"10.1142\/S021819670500261X"},{"key":"S0218196721500454BIB005","doi-asserted-by":"publisher","DOI":"10.1007\/BF01388519"},{"key":"S0218196721500454BIB006","doi-asserted-by":"publisher","DOI":"10.1016\/0022-4049(87)90015-6"},{"key":"S0218196721500454BIB007","doi-asserted-by":"publisher","DOI":"10.1007\/BF01388451"},{"key":"S0218196721500454BIB008","first-page":"215","volume":"42","author":"Cannon J. W.","year":"1996","journal-title":"Enseign. Math. 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