{"id":"https://openalex.org/W2158684550","doi":"https://doi.org/10.1142/s0218196712500166","title":"RELATIVELY HYPERBOLIC GROUPS","display_name":"RELATIVELY HYPERBOLIC GROUPS","publication_year":2011,"publication_date":"2011-10-19","ids":{"openalex":"https://openalex.org/W2158684550","doi":"https://doi.org/10.1142/s0218196712500166","mag":"2158684550"},"language":"en","primary_location":{"id":"doi:10.1142/s0218196712500166","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218196712500166","pdf_url":null,"source":{"id":"https://openalex.org/S23260266","display_name":"International Journal of Algebra and Computation","issn_l":"0218-1967","issn":["0218-1967","1793-6500"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319815","host_organization_name":"World Scientific","host_organization_lineage":["https://openalex.org/P4310319815"],"host_organization_lineage_names":["World Scientific"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Algebra and Computation","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5025232038","display_name":"\u0392. H. Bowditch","orcid":null},"institutions":[{"id":"https://openalex.org/I39555362","display_name":"University of Warwick","ror":"https://ror.org/01a77tt86","country_code":"GB","type":"education","lineage":["https://openalex.org/I39555362"]}],"countries":["GB"],"is_corresponding":true,"raw_author_name":"B. H. BOWDITCH","raw_affiliation_strings":["Mathematics Institute, University of Warwick, Coventry CV4 7AL, Great Britain, UK"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Mathematics Institute, University of Warwick, Coventry CV4 7AL, Great Britain, UK","institution_ids":["https://openalex.org/I39555362"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5025232038"],"corresponding_institution_ids":["https://openalex.org/I39555362"],"apc_list":null,"apc_paid":null,"fwci":26.7051,"has_fulltext":false,"cited_by_count":427,"citation_normalized_percentile":{"value":0.99864098,"is_in_top_1_percent":true,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":99,"max":100},"biblio":{"volume":"22","issue":"03","first_page":"1250016","last_page":"1250016"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10304","display_name":"Geometric and Algebraic Topology","score":1.0,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10304","display_name":"Geometric and Algebraic Topology","score":1.0,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.9894999861717224,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10896","display_name":"Homotopy and Cohomology in Algebraic Topology","score":0.9876000285148621,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.9154738187789917},{"id":"https://openalex.org/keywords/relatively-hyperbolic-group","display_name":"Relatively hyperbolic group","score":0.7243589162826538},{"id":"https://openalex.org/keywords/hyperbolic-group","display_name":"Hyperbolic group","score":0.6454366445541382},{"id":"https://openalex.org/keywords/kleinian-group","display_name":"Kleinian group","score":0.6099739074707031},{"id":"https://openalex.org/keywords/hyperbolic-manifold","display_name":"Hyperbolic manifold","score":0.5749977827072144},{"id":"https://openalex.org/keywords/hyperbolic-space","display_name":"Hyperbolic space","score":0.5390832424163818},{"id":"https://openalex.org/keywords/bounded-function","display_name":"Bounded function","score":0.5250442028045654},{"id":"https://openalex.org/keywords/boundary","display_name":"Boundary (topology)","score":0.5247535705566406},{"id":"https://openalex.org/keywords/conical-surface","display_name":"Conical surface","score":0.509098470211029},{"id":"https://openalex.org/keywords/limit-set","display_name":"Limit set","score":0.4914456605911255},{"id":"https://openalex.org/keywords/group","display_name":"Group (periodic table)","score":0.47495508193969727},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.4601036310195923},{"id":"https://openalex.org/keywords/action","display_name":"Action (physics)","score":0.45057445764541626},{"id":"https://openalex.org/keywords/finitely-generated-group","display_name":"Finitely generated group","score":0.44026854634284973},{"id":"https://openalex.org/keywords/limit","display_name":"Limit (mathematics)","score":0.41628673672676086},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.39187493920326233},{"id":"https://openalex.org/keywords/finitely-generated-abelian-group","display_name":"Finitely-generated abelian group","score":0.20194783806800842},{"id":"https://openalex.org/keywords/hyperbolic-function","display_name":"Hyperbolic function","score":0.1904606819152832},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.15375378727912903}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.9154738187789917},{"id":"https://openalex.org/C40020168","wikidata":"https://www.wikidata.org/wiki/Q7310818","display_name":"Relatively hyperbolic group","level":4,"score":0.7243589162826538},{"id":"https://openalex.org/C167150821","wikidata":"https://www.wikidata.org/wiki/Q81194","display_name":"Hyperbolic group","level":4,"score":0.6454366445541382},{"id":"https://openalex.org/C49527426","wikidata":"https://www.wikidata.org/wiki/Q4116484","display_name":"Kleinian group","level":2,"score":0.6099739074707031},{"id":"https://openalex.org/C70984080","wikidata":"https://www.wikidata.org/wiki/Q2619102","display_name":"Hyperbolic manifold","level":3,"score":0.5749977827072144},{"id":"https://openalex.org/C83677898","wikidata":"https://www.wikidata.org/wiki/Q1878538","display_name":"Hyperbolic space","level":2,"score":0.5390832424163818},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.5250442028045654},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.5247535705566406},{"id":"https://openalex.org/C124961601","wikidata":"https://www.wikidata.org/wiki/Q2518149","display_name":"Conical surface","level":2,"score":0.509098470211029},{"id":"https://openalex.org/C51337361","wikidata":"https://www.wikidata.org/wiki/Q1825444","display_name":"Limit set","level":3,"score":0.4914456605911255},{"id":"https://openalex.org/C2781311116","wikidata":"https://www.wikidata.org/wiki/Q83306","display_name":"Group (periodic table)","level":2,"score":0.47495508193969727},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.4601036310195923},{"id":"https://openalex.org/C2780791683","wikidata":"https://www.wikidata.org/wiki/Q846785","display_name":"Action (physics)","level":2,"score":0.45057445764541626},{"id":"https://openalex.org/C2780811992","wikidata":"https://www.wikidata.org/wiki/Q21083858","display_name":"Finitely generated group","level":3,"score":0.44026854634284973},{"id":"https://openalex.org/C151201525","wikidata":"https://www.wikidata.org/wiki/Q177239","display_name":"Limit (mathematics)","level":2,"score":0.41628673672676086},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.39187493920326233},{"id":"https://openalex.org/C162860070","wikidata":"https://www.wikidata.org/wiki/Q142140","display_name":"Finitely-generated abelian group","level":2,"score":0.20194783806800842},{"id":"https://openalex.org/C92047909","wikidata":"https://www.wikidata.org/wiki/Q204034","display_name":"Hyperbolic function","level":2,"score":0.1904606819152832},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.15375378727912903},{"id":"https://openalex.org/C178790620","wikidata":"https://www.wikidata.org/wiki/Q11351","display_name":"Organic chemistry","level":1,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1142/s0218196712500166","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218196712500166","pdf_url":null,"source":{"id":"https://openalex.org/S23260266","display_name":"International Journal of Algebra and Computation","issn_l":"0218-1967","issn":["0218-1967","1793-6500"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319815","host_organization_name":"World Scientific","host_organization_lineage":["https://openalex.org/P4310319815"],"host_organization_lineage_names":["World Scientific"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"International Journal of Algebra and Computation","raw_type":"journal-article"},{"id":"pmh:oai:eprints.soton.ac.uk:29769","is_oa":false,"landing_page_url":null,"pdf_url":null,"source":{"id":"https://openalex.org/S4306401019","display_name":"ePrints Soton (University of Southampton)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I43439940","host_organization_name":"University of Southampton","host_organization_lineage":["https://openalex.org/I43439940"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"acceptedVersion","is_accepted":true,"is_published":false,"raw_source_name":"","raw_type":"Article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.371.7928","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.371.7928","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://homepages.warwick.ac.uk/~masgak/papers/rhg.pdf","raw_type":"text"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":30,"referenced_works":["https://openalex.org/W1234885875","https://openalex.org/W1490828109","https://openalex.org/W1523420055","https://openalex.org/W1530877435","https://openalex.org/W1578915963","https://openalex.org/W1597188000","https://openalex.org/W1927956358","https://openalex.org/W1963744099","https://openalex.org/W1965377185","https://openalex.org/W1967360215","https://openalex.org/W1975757718","https://openalex.org/W1979060425","https://openalex.org/W1991691624","https://openalex.org/W1997160882","https://openalex.org/W2037070544","https://openalex.org/W2042590978","https://openalex.org/W2046984348","https://openalex.org/W2055527098","https://openalex.org/W2056028905","https://openalex.org/W2058259238","https://openalex.org/W2068638265","https://openalex.org/W2074882953","https://openalex.org/W2084146443","https://openalex.org/W2087965508","https://openalex.org/W2088805036","https://openalex.org/W2102102535","https://openalex.org/W2113006003","https://openalex.org/W2963196760","https://openalex.org/W4206038095","https://openalex.org/W4235484205"],"related_works":["https://openalex.org/W3103116655","https://openalex.org/W2950397580","https://openalex.org/W1557210002","https://openalex.org/W2963819432","https://openalex.org/W2043693710","https://openalex.org/W3091511557","https://openalex.org/W2131647356","https://openalex.org/W2158684550","https://openalex.org/W4365449685","https://openalex.org/W2951772129"],"abstract_inverted_index":{"In":[0],"this":[1,29,115,166],"paper":[2],"we":[3,108],"develop":[4,109],"some":[5,110],"of":[6,9,12,25,28,35,72,111,114,134,143,154,159,172],"the":[7,10,23,112,141,151,160,183],"foundations":[8],"theory":[11],"relatively":[13,121,145],"hyperbolic":[14,48,84,122,132,146],"groups":[15],"as":[16],"originally":[17],"formulated":[18],"by":[19],"Gromov.":[20],"We":[21,86,117,139,179],"prove":[22],"equivalence":[24],"two":[26],"definitions":[27],"notion.":[30,116],"One":[31],"is":[32,57,70,162,185],"essentially":[33],"that":[34,50,53,71,150],"a":[36,39,46,59,64,73,77,81,88,100,120,130,135,144,170],"group":[37,74,123,161],"admitting":[38],"properly":[40],"discontinuous":[41],"geometrically":[42,156,175],"finite":[43,157,176],"action":[44,79,158],"on":[45,80,129],"proper":[47,131],"space,":[49],"is,":[51],"such":[52],"every":[54],"limit":[55,61,152],"point":[56,62],"either":[58],"conical":[60],"or":[63],"bounded":[65],"parabolic":[66],"point.":[67],"The":[68],"other":[69],"which":[75],"admits":[76],"cofinite":[78],"connected":[82],"fine":[83],"graph.":[85],"define":[87,140],"graph":[89],"to":[90,127,165],"be":[91,125],"\"fine\"":[92],"if":[93],"there":[94],"are":[95],"only":[96],"finitely":[97],"many":[98],"circuits":[99],"given":[101,105],"length":[102],"containing":[103],"any":[104,155],"edge,":[106],"and":[107,148],"properties":[113],"show":[118,149],"how":[119],"can":[124],"assumed":[126],"act":[128],"space":[133],"particular":[136],"geometric":[137],"form.":[138],"boundary":[142,184],"group,":[147],"set":[153],"equivariantly":[163],"homeomorphic":[164],"boundary.":[167],"This":[168],"generalizes":[169],"result":[171],"Tukia":[173],"for":[174],"kleinian":[177],"groups.":[178],"also":[180],"describe":[181],"when":[182],"connected.":[186]},"counts_by_year":[{"year":2026,"cited_by_count":13},{"year":2025,"cited_by_count":31},{"year":2024,"cited_by_count":20},{"year":2023,"cited_by_count":23},{"year":2022,"cited_by_count":18},{"year":2021,"cited_by_count":26},{"year":2020,"cited_by_count":32},{"year":2019,"cited_by_count":22},{"year":2018,"cited_by_count":25},{"year":2017,"cited_by_count":24},{"year":2016,"cited_by_count":30},{"year":2015,"cited_by_count":21},{"year":2014,"cited_by_count":14},{"year":2013,"cited_by_count":26},{"year":2012,"cited_by_count":17}],"updated_date":"2026-05-23T08:51:43.019350","created_date":"2025-10-10T00:00:00"}