{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,22]],"date-time":"2025-10-22T18:13:09Z","timestamp":1761156789018},"reference-count":24,"publisher":"Cambridge University Press (CUP)","issue":"3","license":[{"start":{"date-parts":[[2019,12,12]],"date-time":"2019-12-12T00:00:00Z","timestamp":1576108800000},"content-version":"unspecified","delay-in-days":0,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Proc. Camb. Phil. Soc."],"published-print":{"date-parts":[[2021,5]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In this paper we continue the study of right-angled Artin groups up to commensurability initiated in [<jats:bold>CKZ<\/jats:bold>]. We show that RAAGs defined by different paths of length greater than 3 are not commensurable. We also characterise which RAAGs defined by paths are commensurable to RAAGs defined by trees of diameter 4. More precisely, we show that a RAAG defined by a path of length<jats:italic>n<\/jats:italic>&gt; 4 is commensurable to a RAAG defined by a tree of diameter 4 if and only if<jats:italic>n<\/jats:italic>\u2261 2 (mod 4). These results follow from the connection that we establish between the classification of RAAGs up to commensurability and linear integer-programming.<\/jats:p>","DOI":"10.1017\/s0305004119000537","type":"journal-article","created":{"date-parts":[[2019,12,12]],"date-time":"2019-12-12T09:44:17Z","timestamp":1576143857000},"page":"559-608","source":"Crossref","is-referenced-by-count":1,"title":["On commensurability of right-angled Artin groups II: RAAGs defined by paths"],"prefix":"10.1017","volume":"170","author":[{"given":"MONTSERRAT","family":"CASALS\u2013RUIZ","sequence":"first","affiliation":[]},{"given":"ILYA","family":"KAZACHKOV","sequence":"additional","affiliation":[]},{"given":"ALEXANDER","family":"ZAKHAROV","sequence":"additional","affiliation":[]}],"member":"56","published-online":{"date-parts":[[2019,12,12]]},"reference":[{"key":"S0305004119000537_ref24","unstructured":"[Wi96] Wise, D. . Non-Positively Curved Squared Complexes, Aperiodic Tilings and Non-Residually Finite Groups (Princeton University, 1996)."},{"key":"S0305004119000537_ref15","doi-asserted-by":"publisher","DOI":"10.2140\/gt.2013.17.493"},{"key":"S0305004119000537_ref1","doi-asserted-by":"crossref","first-page":"1045","DOI":"10.4171\/dm\/421","article-title":"The virtual Haken conjecture","volume":"18","author":"Agol","journal-title":"Doc. Math"},{"key":"S0305004119000537_ref21","doi-asserted-by":"publisher","DOI":"10.2307\/1970577"},{"key":"S0305004119000537_ref16","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1142\/S021819671450009X","article-title":"The geometry of the curve graph of a right-angled Artin group","volume":"24","author":"Kim","year":"2014","journal-title":"Int. J. Algebra Comput"},{"key":"S0305004119000537_ref20","doi-asserted-by":"publisher","DOI":"10.2307\/2371774"},{"key":"S0305004119000537_ref17","first-page":"88","article-title":"Arithmeticity of nonuniform lattices","volume":"7","author":"Margulis","year":"1973","journal-title":"Funkcional. Anal. i Prilozen."},{"key":"S0305004119000537_ref11","doi-asserted-by":"publisher","DOI":"10.1007\/BF02698687"},{"key":"S0305004119000537_ref9","doi-asserted-by":"publisher","DOI":"10.4171\/GGD\/355"},{"key":"S0305004119000537_ref5","doi-asserted-by":"publisher","DOI":"10.1007\/BF02698916"},{"key":"S0305004119000537_ref14","doi-asserted-by":"publisher","DOI":"10.1017\/S1446788700015445"},{"key":"S0305004119000537_ref18","doi-asserted-by":"publisher","DOI":"10.1007\/BF02698639"},{"key":"S0305004119000537_ref4","doi-asserted-by":"publisher","DOI":"10.1215\/S0012-7094-08-14121-3"},{"key":"S0305004119000537_ref7","unstructured":"[CH17] Chistikov, D. and Haase, C. . On the complexity of quantified integer programming. In ICALP\u201917, vol. 80 of LIPIcs (2017), pp 94:1\u201394:13."},{"key":"S0305004119000537_ref2","doi-asserted-by":"publisher","DOI":"10.1112\/plms\/s3-25.4.603"},{"key":"S0305004119000537_ref13","doi-asserted-by":"publisher","DOI":"10.1007\/s00222-018-0803-3"},{"key":"S0305004119000537_ref8","volume-title":"Annals of Math. Stud","author":"Deligne","year":"1993"},{"key":"S0305004119000537_ref22","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-1962-0133816-6"},{"key":"S0305004119000537_ref19","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-61856-7"},{"key":"S0305004119000537_ref6","doi-asserted-by":"publisher","DOI":"10.4171\/rmi\/1061"},{"key":"S0305004119000537_ref23","unstructured":"[Wh10] Whyte, K. . Coarse bundles, arXiv:1006.3347."},{"key":"S0305004119000537_ref3","doi-asserted-by":"publisher","DOI":"10.1090\/S0002-9939-08-09559-2"},{"key":"S0305004119000537_ref12","first-page":"1","article-title":"Geometric group theory, vol. 2: asymptotic invariants of infinite groups","volume":"182","author":"Gromov","year":"1993","journal-title":"London Math. Soc. 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