{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,6,13]],"date-time":"2025-06-13T04:03:58Z","timestamp":1749787438659,"version":"3.41.0"},"reference-count":20,"publisher":"World Scientific Pub Co Pte Ltd","issue":"04","funder":[{"name":"National Science Foundation","award":["DMS-2202942"],"award-info":[{"award-number":["DMS-2202942"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Algebra Comput."],"published-print":{"date-parts":[[2025,6]]},"abstract":"<jats:p> Generalizing Culler and Vogtmann\u2019s Outer Space for the free group, Guirardel and Levitt construct an Outer Space for a free product of groups. We completely characterize when this space (or really its simplicial spine) supports an equivariant piecewise-Euclidean or piecewise-hyperbolic CAT(0) metric. Our results are mostly negative, extending thesis work of Bridson and related to thesis work of Cunningham. In particular, provided the dimension of the spine is at least three, it is never CAT(0). Surprisingly, we exhibit one family of free products for which the Outer Space is two-dimensional and does support an equivariant CAT(0) metric. <\/jats:p>","DOI":"10.1142\/s0218196725500134","type":"journal-article","created":{"date-parts":[[2025,3,14]],"date-time":"2025-03-14T10:14:01Z","timestamp":1741947241000},"page":"505-536","source":"Crossref","is-referenced-by-count":0,"title":["When is the Outer Space of a free product CAT(0)?"],"prefix":"10.1142","volume":"35","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8083-4747","authenticated-orcid":false,"given":"Robert Alonzo","family":"Lyman","sequence":"first","affiliation":[{"name":"Rutgers University\u2013Newark, Department of Mathematics and Computer Science, Smith Hall 101 Warren St, Newark, NJ 07102, United States"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"219","published-online":{"date-parts":[[2025,4,25]]},"reference":[{"key":"S0218196725500134BIB001","doi-asserted-by":"publisher","DOI":"10.1016\/0022-4049(93)90085-8"},{"key":"S0218196725500134BIB002","doi-asserted-by":"publisher","DOI":"10.1007\/s00208-008-0317-1"},{"key":"S0218196725500134BIB003","doi-asserted-by":"publisher","DOI":"10.1007\/s002220000068"},{"key":"S0218196725500134BIB004","first-page":"373","volume-title":"Group Theory From a Geometrical Viewpoint (Trieste, 1990)","author":"Bridson M. 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Miller,  Symmetric Automorphisms of Free Products,  Memoirs of the American Mathematical Society, Vol.  582  (American Mathematical Society,  1996),  viii+97 pp.","DOI":"10.1090\/memo\/0582"},{"key":"S0218196725500134BIB020","series-title":"Springer Monographs in Mathematics","volume-title":"Trees","author":"Serre J.-P.","year":"2003"},{"key":"S0218196725500134BIB021","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781107325449.007"},{"key":"S0218196725500134BIB022","doi-asserted-by":"publisher","DOI":"10.1016\/0040-9383(94)00042-J"}],"container-title":["International Journal of Algebra and Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218196725500134","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,12]],"date-time":"2025-06-12T07:50:20Z","timestamp":1749714620000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/10.1142\/S0218196725500134"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,4,25]]},"references-count":20,"journal-issue":{"issue":"04","published-print":{"date-parts":[[2025,6]]}},"alternative-id":["10.1142\/S0218196725500134"],"URL":"https:\/\/doi.org\/10.1142\/s0218196725500134","relation":{},"ISSN":["0218-1967","1793-6500"],"issn-type":[{"type":"print","value":"0218-1967"},{"type":"electronic","value":"1793-6500"}],"subject":[],"published":{"date-parts":[[2025,4,25]]}}}