{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:RQ4HL7VZGP5VKFYNJPGXAPWGM5","short_pith_number":"pith:RQ4HL7VZ","schema_version":"1.0","canonical_sha256":"8c3875feb933fb55170d4bcd703ec6675288b741de76b573d2aa644197dc743e","source":{"kind":"arxiv","id":"1606.03994","version":2},"attestation_state":"computed","paper":{"title":"On the large-scale geometry of diffeomorphism groups of $1$-manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Michael P. Cohen","submitted_at":"2016-06-13T15:33:17Z","abstract_excerpt":"We apply the framework of Rosendal to study the large-scale geometry of the topological groups $\\Diff_+^k(M^1)$, consisting of orientation-preserving $C^k$-diffeomorphisms (for $1\\leq k\\leq\\infty$) of a compact $1$-manifold $M^1$ ($=I$ or $\\mathbb{S}^1$). We characterize the relative property (OB) in such groups: $A\\subseteq\\Diff_+^k(M^1)$ has property (OB) relative to $\\Diff_+^k(M^1)$ if and only if $\\displaystyle\\sup_{f\\in A}\\sup_{x\\in M^1}|\\log f'(x)|<\\infty$ and $\\displaystyle\\sup_{f\\in A}\\sup_{x\\in M^1}|f^{(j)}(x)|<\\infty$ for every integer $2\\leq j\\leq k$. We deduce that $\\Diff_+^k(M^1)$"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1606.03994","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-06-13T15:33:17Z","cross_cats_sorted":[],"title_canon_sha256":"aa247f62c4919e0fa8e7e77116030c6ce456ac68b22a7c35e5a90afaddeb9836","abstract_canon_sha256":"f27226914d78ce125763253c8f7a34f7b7991c308ff71b4207e04f862e3bc19d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:49:37.922449Z","signature_b64":"Lbfc/USk8X+ubOKsMahyHJi/uW8DR6+2AYJ9Tj6yaA/Z4kSs6jzWVlJgURobNLVQrkCnlBDszMUm29Zrdso8Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8c3875feb933fb55170d4bcd703ec6675288b741de76b573d2aa644197dc743e","last_reissued_at":"2026-05-18T00:49:37.921759Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:49:37.921759Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the large-scale geometry of diffeomorphism groups of $1$-manifolds","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Michael P. Cohen","submitted_at":"2016-06-13T15:33:17Z","abstract_excerpt":"We apply the framework of Rosendal to study the large-scale geometry of the topological groups $\\Diff_+^k(M^1)$, consisting of orientation-preserving $C^k$-diffeomorphisms (for $1\\leq k\\leq\\infty$) of a compact $1$-manifold $M^1$ ($=I$ or $\\mathbb{S}^1$). We characterize the relative property (OB) in such groups: $A\\subseteq\\Diff_+^k(M^1)$ has property (OB) relative to $\\Diff_+^k(M^1)$ if and only if $\\displaystyle\\sup_{f\\in A}\\sup_{x\\in M^1}|\\log f'(x)|<\\infty$ and $\\displaystyle\\sup_{f\\in A}\\sup_{x\\in M^1}|f^{(j)}(x)|<\\infty$ for every integer $2\\leq j\\leq k$. We deduce that $\\Diff_+^k(M^1)$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.03994","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1606.03994","created_at":"2026-05-18T00:49:37.921842+00:00"},{"alias_kind":"arxiv_version","alias_value":"1606.03994v2","created_at":"2026-05-18T00:49:37.921842+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1606.03994","created_at":"2026-05-18T00:49:37.921842+00:00"},{"alias_kind":"pith_short_12","alias_value":"RQ4HL7VZGP5V","created_at":"2026-05-18T12:30:41.710351+00:00"},{"alias_kind":"pith_short_16","alias_value":"RQ4HL7VZGP5VKFYN","created_at":"2026-05-18T12:30:41.710351+00:00"},{"alias_kind":"pith_short_8","alias_value":"RQ4HL7VZ","created_at":"2026-05-18T12:30:41.710351+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/RQ4HL7VZGP5VKFYNJPGXAPWGM5","json":"https://pith.science/pith/RQ4HL7VZGP5VKFYNJPGXAPWGM5.json","graph_json":"https://pith.science/api/pith-number/RQ4HL7VZGP5VKFYNJPGXAPWGM5/graph.json","events_json":"https://pith.science/api/pith-number/RQ4HL7VZGP5VKFYNJPGXAPWGM5/events.json","paper":"https://pith.science/paper/RQ4HL7VZ"},"agent_actions":{"view_html":"https://pith.science/pith/RQ4HL7VZGP5VKFYNJPGXAPWGM5","download_json":"https://pith.science/pith/RQ4HL7VZGP5VKFYNJPGXAPWGM5.json","view_paper":"https://pith.science/paper/RQ4HL7VZ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1606.03994&json=true","fetch_graph":"https://pith.science/api/pith-number/RQ4HL7VZGP5VKFYNJPGXAPWGM5/graph.json","fetch_events":"https://pith.science/api/pith-number/RQ4HL7VZGP5VKFYNJPGXAPWGM5/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/RQ4HL7VZGP5VKFYNJPGXAPWGM5/action/timestamp_anchor","attest_storage":"https://pith.science/pith/RQ4HL7VZGP5VKFYNJPGXAPWGM5/action/storage_attestation","attest_author":"https://pith.science/pith/RQ4HL7VZGP5VKFYNJPGXAPWGM5/action/author_attestation","sign_citation":"https://pith.science/pith/RQ4HL7VZGP5VKFYNJPGXAPWGM5/action/citation_signature","submit_replication":"https://pith.science/pith/RQ4HL7VZGP5VKFYNJPGXAPWGM5/action/replication_record"}},"created_at":"2026-05-18T00:49:37.921842+00:00","updated_at":"2026-05-18T00:49:37.921842+00:00"}