{"id":"https://openalex.org/W2031411620","doi":"https://doi.org/10.1142/s0218196701000565","title":"ON RANK, ROOT AND EQUATIONS IN FREE GROUPS","display_name":"ON RANK, ROOT AND EQUATIONS IN FREE GROUPS","publication_year":2001,"publication_date":"2001-06-01","ids":{"openalex":"https://openalex.org/W2031411620","doi":"https://doi.org/10.1142/s0218196701000565","mag":"2031411620"},"language":"en","primary_location":{"id":"doi:10.1142/s0218196701000565","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218196701000565","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/A5014949895","display_name":"Amnon Rosenmann","orcid":"https://orcid.org/0000-0003-0255-0885"},"institutions":[{"id":"https://openalex.org/I16391192","display_name":"Tel Aviv University","ror":"https://ror.org/04mhzgx49","country_code":"IL","type":"education","lineage":["https://openalex.org/I16391192"]}],"countries":["IL"],"is_corresponding":true,"raw_author_name":"AMNON ROSENMANN","raw_affiliation_strings":["School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel","School of Mathematical Sciences, Tel-Aviv University, Ramat-Aviv, Tel-Aviv 69978, Israel ("],"affiliations":[{"raw_affiliation_string":"School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel","institution_ids":["https://openalex.org/I16391192"]},{"raw_affiliation_string":"School of Mathematical Sciences, Tel-Aviv University, Ramat-Aviv, Tel-Aviv 69978, Israel (","institution_ids":["https://openalex.org/I16391192"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5014949895"],"corresponding_institution_ids":["https://openalex.org/I16391192"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":4,"citation_normalized_percentile":{"value":0.15046492,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"11","issue":"03","first_page":"375","last_page":"390"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10304","display_name":"Geometric and Algebraic Topology","score":0.9998999834060669,"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":0.9998999834060669,"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/T10899","display_name":"Advanced Operator Algebra Research","score":0.9983000159263611,"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":0.996399998664856,"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":0.8917793035507202},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.7941811084747314},{"id":"https://openalex.org/keywords/rank","display_name":"Rank (graph theory)","score":0.7879482507705688},{"id":"https://openalex.org/keywords/free-group","display_name":"Free group","score":0.5663394331932068},{"id":"https://openalex.org/keywords/sequence","display_name":"Sequence (biology)","score":0.49510374665260315},{"id":"https://openalex.org/keywords/corollary","display_name":"Corollary","score":0.47480928897857666},{"id":"https://openalex.org/keywords/group","display_name":"Group (periodic table)","score":0.4541579484939575},{"id":"https://openalex.org/keywords/word","display_name":"Word (group theory)","score":0.44612210988998413},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.32388967275619507}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8917793035507202},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7941811084747314},{"id":"https://openalex.org/C164226766","wikidata":"https://www.wikidata.org/wiki/Q7293202","display_name":"Rank (graph theory)","level":2,"score":0.7879482507705688},{"id":"https://openalex.org/C77021192","wikidata":"https://www.wikidata.org/wiki/Q431078","display_name":"Free group","level":3,"score":0.5663394331932068},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.49510374665260315},{"id":"https://openalex.org/C2780012671","wikidata":"https://www.wikidata.org/wiki/Q1343870","display_name":"Corollary","level":2,"score":0.47480928897857666},{"id":"https://openalex.org/C2781311116","wikidata":"https://www.wikidata.org/wiki/Q83306","display_name":"Group (periodic table)","level":2,"score":0.4541579484939575},{"id":"https://openalex.org/C90805587","wikidata":"https://www.wikidata.org/wiki/Q10944557","display_name":"Word (group theory)","level":2,"score":0.44612210988998413},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.32388967275619507},{"id":"https://openalex.org/C54355233","wikidata":"https://www.wikidata.org/wiki/Q7162","display_name":"Genetics","level":1,"score":0.0},{"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/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","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/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1142/s0218196701000565","is_oa":false,"landing_page_url":"https://doi.org/10.1142/s0218196701000565","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"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":10,"referenced_works":["https://openalex.org/W1523480252","https://openalex.org/W1595661776","https://openalex.org/W2000922387","https://openalex.org/W2011199327","https://openalex.org/W2013923794","https://openalex.org/W2017370604","https://openalex.org/W2018996603","https://openalex.org/W2037835994","https://openalex.org/W2073032234","https://openalex.org/W2325546988"],"related_works":["https://openalex.org/W2044496651","https://openalex.org/W3008193251","https://openalex.org/W2088225562","https://openalex.org/W2160600548","https://openalex.org/W2039508012","https://openalex.org/W3089291528","https://openalex.org/W2962915575","https://openalex.org/W4288642074","https://openalex.org/W2907043423","https://openalex.org/W2066605111"],"abstract_inverted_index":{"Let":[0,24],"h":[1,4,46,51],"1":[2,32,112,189],",":[3,33,36,53,110,113,116],"2":[5,35,115],",\u2026":[6],"be":[7,19,26],"a":[8,13,43,73,151,162,184],"sequence":[9,105,124],"of":[10,92,106,142,153,161],"elements":[11,94],"in":[12,45,95,144,176],"free":[14,145],"group":[15],"and":[16,48,103],"let":[17],"H":[18,98,108,111,114,130],"the":[20,27,56,61,86,90,93,104,123,154,158,182,194],"subgroup":[21,28,87],"they":[22],"generate.":[23],"H\u2032":[25],"generated":[29,88,102],"by":[30,89],"w":[31,34,40],"\u2026,":[37],"where":[38,81],"each":[39],"i":[41,47,131,185],"is":[42,85,100,197],"word":[44],"possibly":[49],"other":[50],"j":[52],"such":[54,156],"that":[55,67,77,157,174],"associated":[57],"directed":[58],"graph":[59],"has":[60],"finite":[62],"paths":[63],"property.":[64],"We":[65],"show":[66,173],"rank":[68,70,160,196],"H\u2032\u2265":[69],"H.":[71,96],"As":[72],"corollary":[74],"we":[75,147,172],"get":[76],"[Formula:":[78,82,119,179],"see":[79,83,120,180],"text],":[80],"text]":[84,121,181],"roots":[91],"If":[97],"0":[99,109],"finitely":[101],"subgroups":[107],"\u2026":[117],"satisfies":[118],"then":[122,193],"stabilizes,":[125],"i.e.":[126],"for":[127,134],"some":[128],"m,":[129],"=H":[132],"i+1":[133],"every":[135],"i\u2265":[136],"m.":[137],"When":[138],"applied":[139],"to":[140],"systems":[141],"equations":[143],"groups,":[146],"give":[148],"conditions":[149],"on":[150],"transformation":[152],"system":[155],"maximal":[159],"solution":[163],"(the":[164],"inner":[165,195],"rank)":[166],"does":[167],"not":[168],"increase.":[169],"In":[170],"particular,":[171],"if":[175],"\"Lyndon":[177],"equation\"":[178],"exponents":[183],"satisfy":[186],"gcd":[187],"(a":[188],",\u2026,a":[190],"n":[191],")\u22601":[192],"\u230a":[198],"n/2\u230b.":[199],"The":[200],"proofs":[201],"are":[202],"mostly":[203],"elementary.":[204]},"counts_by_year":[{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":2},{"year":2021,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}