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Thus if the groupoid is also ample and minimal, it satisfies Matui's AH conjecture.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The polynomial growth condition on the groupoid, together with second-countability and the étale property, is assumed to be sufficient to construct sequences witnessing topological amenability (or weak m-comparison) without additional restrictions on the groupoid structure.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Second-countable étale groupoids with polynomial growth are topologically amenable, and under compactness, metrizability, ampleness and minimality they satisfy weak m-comparison and Matui's AH conjecture.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Second-countable étale groupoids with polynomial growth are topologically amenable.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"f856a4789d18ccd814c5df97b1a917f0fbb7413e6e25107b72db1e2bcfc7fd41"},"source":{"id":"2605.16013","kind":"arxiv","version":1},"verdict":{"id":"ce6100dc-6112-48b3-bcce-7b290b0c8898","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T18:51:25.704264Z","strongest_claim":"We show that any second-countable étale groupoid with polynomial growth is topologically amenable. 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Thus if the groupoid is also ample and minimal, it satisfies Matui's AH conjecture.","one_line_summary":"Second-countable étale groupoids with polynomial growth are topologically amenable, and under compactness, metrizability, ampleness and minimality they satisfy weak m-comparison and Matui's AH conjecture.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The polynomial growth condition on the groupoid, together with second-countability and the étale property, is assumed to be sufficient to construct sequences witnessing topological amenability (or weak m-comparison) without additional restrictions on the groupoid structure.","pith_extraction_headline":"Second-countable étale groupoids with polynomial growth are topologically amenable."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.16013/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_compliance","ran_at":"2026-05-19T19:01:34.295673Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T19:01:18.992187Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T17:33:42.164085Z","status":"skipped","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T17:01:55.642762Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"edae0f68b39d61c2028016d7770ed9a84a9a655e561bb2817166422b1e1ad879"},"references":{"count":44,"sample":[{"doi":"","year":2016,"title":"2016 , Eprint =","work_id":"a74234e6-dbf2-41c9-a2e9-98523203b5e2","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Fiberwise amenability of ample \\'","work_id":"656f1071-b62a-4d4e-8cfb-52e09b49e055","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1090/surv/241","year":2019,"title":"Williams, Dana P. , title =. 2019 , publisher =. doi:10.1090/surv/241 , keywords =","work_id":"af033f1d-4e3e-49ae-800a-a06c568c4f0a","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2000,"title":"Anantharaman-Delaroche, C. and Renault, J. , TITLE =. 2000 , PAGES =","work_id":"da8636df-f5f8-4d2a-9cab-17cf3664502f","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2019,"title":"On a class of singular measures satisfying a strong annular decay condition , JOURNAL =","work_id":"6b61783e-8fa5-4b1a-a86e-8e2337da67c2","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":44,"snapshot_sha256":"47e8b101a77c1252ceec439873106967567107012fd6a0adfa3eb14f5fd90c19","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"ca138ec86756d26c54f7fec5a842e809d43a54b25c504e3890d51b82c2dbcb2c"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}