{"id":"https://openalex.org/W7214034803","doi":"https://doi.org/10.48550/arxiv.2609.23702","title":"Comparison and Almost Finiteness for Actions of Amenable Groups","display_name":"Comparison and Almost Finiteness for Actions of Amenable Groups","publication_year":2026,"publication_date":"2026-09-20","ids":{"openalex":"https://openalex.org/W7214034803","doi":"https://doi.org/10.48550/arxiv.2609.23702"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.23702","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.23702","pdf_url":"https://arxiv.org/pdf/2609.23702","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/2609.23702","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5047824959","display_name":"Eli Glasner","orcid":"https://orcid.org/0000-0003-1167-1283","observed_orcids":["https://orcid.org/0000-0003-1167-1283"]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Glasner, Eli","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5152482524","display_name":"Chunlin Liu","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Liu, Chunlin","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":4,"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/T10899","display_name":"Advanced Operator Algebra Research","score":0.9082000255584717,"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"}},"topics":[{"id":"https://openalex.org/T10899","display_name":"Advanced Operator Algebra Research","score":0.9082000255584717,"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/T11151","display_name":"Advanced Topology and Set Theory","score":0.03060000017285347,"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":0.025100000202655792,"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/amenable-group-actions","display_name":"amenable group actions","score":0.8500000238418579},{"id":"https://openalex.org/keywords/topological-full-groups","display_name":"topological full groups","score":0.8610000014305115},{"id":"https://openalex.org/keywords/reduced-crossed-products","display_name":"reduced crossed products","score":0.7440000176429749},{"id":"https://openalex.org/keywords/z-stability","display_name":"Z-stability","score":0.9700000286102295},{"id":"https://openalex.org/keywords/nuclear-dimension","display_name":"nuclear dimension","score":0.8709999918937683},{"id":"https://openalex.org/keywords/invariant-probability-measures","display_name":"invariant probability measures","score":0.7089999914169312}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.9000999927520752},{"id":"https://openalex.org/C105433845","wikidata":"https://www.wikidata.org/wiki/Q1194053","display_name":"Metrization theorem","level":3,"score":0.6863999962806702},{"id":"https://openalex.org/C36605576","wikidata":"https://www.wikidata.org/wiki/Q273188","display_name":"Cantor set","level":2,"score":0.6370000243186951},{"id":"https://openalex.org/C207405024","wikidata":"https://www.wikidata.org/wiki/Q207348","display_name":"Semigroup","level":2,"score":0.46149998903274536},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.4607999920845032},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.4260999858379364},{"id":"https://openalex.org/C191399826","wikidata":"https://www.wikidata.org/wiki/Q326908","display_name":"Hausdorff space","level":2,"score":0.4221000075340271},{"id":"https://openalex.org/C190470478","wikidata":"https://www.wikidata.org/wiki/Q2370229","display_name":"Invariant (physics)","level":2,"score":0.414000004529953},{"id":"https://openalex.org/C31498916","wikidata":"https://www.wikidata.org/wiki/Q583034","display_name":"Locally compact space","level":2,"score":0.3732999861240387},{"id":"https://openalex.org/C2781311116","wikidata":"https://www.wikidata.org/wiki/Q83306","display_name":"Group (periodic table)","level":2,"score":0.3634999990463257},{"id":"https://openalex.org/C149701218","wikidata":"https://www.wikidata.org/wiki/Q938883","display_name":"Cantor function","level":3,"score":0.3422999978065491},{"id":"https://openalex.org/C10728891","wikidata":"https://www.wikidata.org/wiki/Q584521","display_name":"Homeomorphism (graph theory)","level":2,"score":0.34139999747276306},{"id":"https://openalex.org/C70710897","wikidata":"https://www.wikidata.org/wiki/Q680081","display_name":"Separable space","level":2,"score":0.33719998598098755},{"id":"https://openalex.org/C154804608","wikidata":"https://www.wikidata.org/wiki/Q2721559","display_name":"Topological manifold","level":5,"score":0.335099995136261},{"id":"https://openalex.org/C21031990","wikidata":"https://www.wikidata.org/wiki/Q355020","display_name":"Probability measure","level":2,"score":0.31520000100135803},{"id":"https://openalex.org/C110729354","wikidata":"https://www.wikidata.org/wiki/Q185478","display_name":"Countable set","level":2,"score":0.3147999942302704},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.3086000084877014},{"id":"https://openalex.org/C17327563","wikidata":"https://www.wikidata.org/wiki/Q1991405","display_name":"Nowhere dense set","level":3,"score":0.3082999885082245},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.298799991607666},{"id":"https://openalex.org/C92757383","wikidata":"https://www.wikidata.org/wiki/Q382497","display_name":"Affine transformation","level":2,"score":0.2930000126361847},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.28189998865127563},{"id":"https://openalex.org/C46467970","wikidata":"https://www.wikidata.org/wiki/Q1361055","display_name":"Discrete group","level":3,"score":0.27720001339912415},{"id":"https://openalex.org/C2777299769","wikidata":"https://www.wikidata.org/wiki/Q3707858","display_name":"Type (biology)","level":2,"score":0.2662000060081482},{"id":"https://openalex.org/C50555996","wikidata":"https://www.wikidata.org/wiki/Q840023","display_name":"Nilpotent","level":2,"score":0.2581999897956848},{"id":"https://openalex.org/C131715654","wikidata":"https://www.wikidata.org/wiki/Q288465","display_name":"Group action","level":3,"score":0.2531999945640564},{"id":"https://openalex.org/C130133044","wikidata":"https://www.wikidata.org/wiki/Q1887083","display_name":"Compact group","level":3,"score":0.25279998779296875}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.23702","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.23702","pdf_url":"https://arxiv.org/pdf/2609.23702","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.2609.23702","doi":"https://doi.org/10.48550/arxiv.2609.23702","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.23702","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:2609.23702","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.23702","pdf_url":"https://arxiv.org/pdf/2609.23702","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":[{"id":"https://openalex.org/G5922688044","display_name":null,"funder_award_id":"2025M773074","funder_id":"https://openalex.org/F4320321543","funder_display_name":"China Postdoctoral Science Foundation"},{"id":"https://openalex.org/G7282035001","display_name":null,"funder_award_id":"BX20250067","funder_id":"https://openalex.org/F4320321543","funder_display_name":"China Postdoctoral Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320321543","display_name":"China Postdoctoral Science Foundation","ror":"https://ror.org/0426zh255"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7214034803.pdf","grobid_xml":"https://content.openalex.org/works/W7214034803.grobid-xml"},"referenced_works_count":2,"referenced_works":["https://openalex.org/W7167215179","https://openalex.org/W7203938472"],"related_works":["https://openalex.org/W4407105445","https://openalex.org/W2771874378","https://openalex.org/W4405190990","https://openalex.org/W2951151419","https://openalex.org/W7167215179","https://openalex.org/W4387542136","https://openalex.org/W2792361732","https://openalex.org/W2243919774","https://openalex.org/W3178701323","https://openalex.org/W3208322908"],"abstract_inverted_index":{"We":[0,123],"prove":[1,99,176],"that":[2,177],"every":[3,145,178],"action":[4,180,215],"of":[5,69,76,89,101,112,156,171,181],"a":[6,13,109,182,189],"countably":[7,183],"infinite":[8,184],"discrete":[9,185],"amenable":[10,87,186],"group":[11,72,187],"on":[12,188],"nonempty":[14,146,190],"compact":[15,191,210],"Hausdorff":[16],"zero-dimensional":[17],"space":[18,116,193],"has":[19,226],"dynamical":[20],"comparison,":[21],"without":[22],"assuming":[23],"minimality,":[24],"freeness,":[25],"or":[26],"metrizability.":[27],"More":[28],"precisely,":[29],"strict":[30],"inequalities":[31],"under":[32],"all":[33,234],"invariant":[34,120],"probability":[35,121],"measures":[36],"imply":[37],"comparison":[38,91,167],"in":[39,238],"the":[40,52,66,70,86,102,113,118,125,141,153,160,169,195,214,235,239],"clopen":[41,53,130,148,157],"type":[42,54],"semigroup":[43,55],"with":[44,117,168,194],"an":[45,74],"order-unit":[46],"remainder.":[47],"For":[48,93],"minimal":[49,94,110],"Cantor":[50,95,115],"actions,":[51,96],"is":[56,73,138,200,216,223],"cancellative":[57],"and":[58,61,85,107,173,232],"almost":[59,201],"unperforated,":[60],"its":[62,219],"canonical":[63],"map":[64,106],"onto":[65],"positive":[67],"cone":[68],"coinvariant":[71],"isomorphism":[75],"ordered":[77],"monoids.":[78],"These":[79],"results":[80],"answer":[81],"Melleray's":[82],"cancellation":[83],"question":[84],"case":[88],"his":[90],"question.":[92],"we":[97,175],"also":[98,217],"goodness":[100],"associated":[103],"affine":[104,133,162],"evaluation":[105,134,163],"obtain":[108],"homeomorphism":[111],"same":[114,119,161],"measures.":[122],"identify":[124],"precise":[126],"obstruction":[127,137],"to":[128],"recovering":[129],"equidecomposability":[131],"from":[132],"alone.":[135],"This":[136,203],"given":[139],"by":[140],"infinitesimal":[142],"subgroup:":[143],"for":[144,208],"proper":[147],"set":[149],"$A$,":[150],"it":[151],"parametrizes":[152],"topological-full-group":[154],"orbits":[155],"sets":[158],"having":[159],"as":[164],"$A$.":[165],"Combining":[166],"theorems":[170],"Kerr":[172],"Szab\u00f3,":[174],"free":[179],"metrizable":[192,211],"topological":[196],"small":[197],"boundary":[198],"property":[199],"finite.":[202],"settles":[204],"Naryshkin's":[205],"almost-finiteness":[206],"conjecture":[207],"finite-dimensional":[209],"spaces.":[212],"If":[213],"minimal,":[218],"reduced":[220],"crossed":[221],"product":[222],"$\\mathcal":[224],"Z$-stable,":[225],"nuclear":[227],"dimension":[228],"at":[229],"most":[230],"one,":[231],"satisfies":[233],"regularity":[236],"conditions":[237],"Toms--Winter":[240],"conjecture.":[241]},"counts_by_year":[{"year":2026,"cited_by_count":4}],"updated_date":"2026-11-08T12:12:28.789561","created_date":"2026-09-23T00:00:00"}
