{"id":"https://openalex.org/W7168290679","doi":"https://doi.org/10.48550/arxiv.2607.11194","title":"On Small Doubling in Right-Ordered Groups and Baumslag-Solitar Groups-II","display_name":"On Small Doubling in Right-Ordered Groups and Baumslag-Solitar Groups-II","publication_year":2026,"publication_date":"2026-07-13","ids":{"openalex":"https://openalex.org/W7168290679","doi":"https://doi.org/10.48550/arxiv.2607.11194"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2607.11194","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2607.11194","pdf_url":"https://arxiv.org/pdf/2607.11194","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2607.11194","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5125189124","display_name":"Mohan","orcid":"https://orcid.org/0009-0006-8355-777X","observed_orcids":["https://orcid.org/0009-0006-8355-777X"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Mohan","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5153880082","display_name":"Neetu","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Neetu","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10849","display_name":"Finite Group Theory Research","score":0.6352999806404114,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"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/T10849","display_name":"Finite Group Theory Research","score":0.6352999806404114,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"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/T11329","display_name":"Limits and Structures in Graph Theory","score":0.062300000339746475,"subfield":{"id":"https://openalex.org/subfields/2607","display_name":"Discrete Mathematics and Combinatorics"},"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/T10304","display_name":"Geometric and Algebraic Topology","score":0.05790000036358833,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/small-doubling","display_name":"small doubling","score":0.9610000252723694},{"id":"https://openalex.org/keywords/baumslag-solitar-groups","display_name":"Baumslag-Solitar groups","score":0.9940000176429749},{"id":"https://openalex.org/keywords/finite-subsets","display_name":"finite subsets","score":0.7260000109672546},{"id":"https://openalex.org/keywords/identity-element","display_name":"identity element","score":0.4779999852180481}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8084999918937683},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.6980000138282776},{"id":"https://openalex.org/C2780990831","wikidata":"https://www.wikidata.org/wiki/Q319141","display_name":"Conjecture","level":2,"score":0.6718000173568726},{"id":"https://openalex.org/C2781311116","wikidata":"https://www.wikidata.org/wiki/Q83306","display_name":"Group (periodic table)","level":2,"score":0.5680000185966492},{"id":"https://openalex.org/C2780841128","wikidata":"https://www.wikidata.org/wiki/Q5073781","display_name":"Characterization (materials science)","level":2,"score":0.4749000072479248},{"id":"https://openalex.org/C2778355321","wikidata":"https://www.wikidata.org/wiki/Q17079427","display_name":"Identity (music)","level":2,"score":0.4426000118255615},{"id":"https://openalex.org/C200288055","wikidata":"https://www.wikidata.org/wiki/Q2621792","display_name":"Element (criminal law)","level":2,"score":0.3815999925136566},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.36980000138282776},{"id":"https://openalex.org/C196773441","wikidata":"https://www.wikidata.org/wiki/Q1475294","display_name":"Maximal element","level":3,"score":0.33329999446868896},{"id":"https://openalex.org/C2777404646","wikidata":"https://www.wikidata.org/wiki/Q1057968","display_name":"Finite group","level":3,"score":0.32749998569488525},{"id":"https://openalex.org/C182306322","wikidata":"https://www.wikidata.org/wiki/Q1779371","display_name":"Order (exchange)","level":2,"score":0.31369999051094055},{"id":"https://openalex.org/C81651864","wikidata":"https://www.wikidata.org/wiki/Q874429","display_name":"Group theory","level":2,"score":0.29899999499320984},{"id":"https://openalex.org/C2777212361","wikidata":"https://www.wikidata.org/wiki/Q5127848","display_name":"Class (philosophy)","level":2,"score":0.2685000002384186},{"id":"https://openalex.org/C177264268","wikidata":"https://www.wikidata.org/wiki/Q1514741","display_name":"Set (abstract data type)","level":2,"score":0.2671999931335449}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2607.11194","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2607.11194","pdf_url":"https://arxiv.org/pdf/2607.11194","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2607.11194","doi":"https://doi.org/10.48550/arxiv.2607.11194","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2607.11194","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2607.11194","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2607.11194","pdf_url":"https://arxiv.org/pdf/2607.11194","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7168290679.pdf","grobid_xml":"https://content.openalex.org/works/W7168290679.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W4417048164","https://openalex.org/W1991900041","https://openalex.org/W2187702617","https://openalex.org/W4410765866","https://openalex.org/W2967093289","https://openalex.org/W2111433766","https://openalex.org/W1649865347","https://openalex.org/W4392224297","https://openalex.org/W2030120764","https://openalex.org/W2783186787"],"abstract_inverted_index":{"Recently,":[0],"Mohan":[1],"et":[2],"al.":[3],"[Results":[4],"Math.":[5],"80":[6],"(2025),":[7],"No.":[8],"4,":[9],"122]":[10],"answered":[11],"Freiman's":[12],"$3k-4$":[13],"conjecture":[14],"in":[15],"right-ordered":[16,39],"groups":[17],"under":[18],"certain":[19],"restrictions.":[20],"In":[21],"this":[22],"paper,":[23],"we":[24,50],"take":[25],"a":[26,38,52],"step":[27],"further":[28],"by":[29],"investigating":[30],"the":[31,42,62,80,84],"structure":[32],"of":[33,37,55,61,86],"nonempty":[34,57],"subsets":[35,59],"$S$":[36,60],"group":[40,64],"satisfying":[41],"small":[43],"doubling":[44],"condition":[45],"$|S^2|":[46,76],"=":[47,77],"3|S|-3$.":[48],"Moreover,":[49],"provide":[51],"complete":[53],"characterization":[54],"all":[56],"finite":[58],"Baumslag-Solitar":[63],"$\\mathrm{BS}(1,q)$":[65],"(with":[66],"$q":[67,71],"\\in":[68],"\\mathbb{Z}$":[69],"and":[70,79],"\\neq":[72],"-1$)":[73],"for":[74],"which":[75],"3|S|-3$":[78],"identity":[81],"element":[82],"is":[83],"minimum":[85],"$S$.":[87]},"counts_by_year":[],"updated_date":"2026-10-28T10:33:24.094632","created_date":"2026-07-15T00:00:00"}
