{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:WMHMVIK2DGAW774CGGD4D4WEV6","short_pith_number":"pith:WMHMVIK2","schema_version":"1.0","canonical_sha256":"b30ecaa15a19816fff823187c1f2c4af88bc76d2dd54bd9be84f00c826a8042a","source":{"kind":"arxiv","id":"1509.00632","version":4},"attestation_state":"computed","paper":{"title":"Hierarchically hyperbolic spaces II: Combination theorems and the distance formula","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Alessandro Sisto, Jason Behrstock, Mark F. Hagen","submitted_at":"2015-09-02T10:34:37Z","abstract_excerpt":"We introduce a number of tools for finding and studying \\emph{hierarchically hyperbolic spaces (HHS)}, a rich class of spaces including mapping class groups of surfaces, Teichm\\\"{u}ller space with either the Teichm\\\"{u}ller or Weil-Petersson metrics, right-angled Artin groups, and the universal cover of any compact special cube complex. We begin by introducing a streamlined set of axioms defining an HHS. We prove that all HHSs satisfy a Masur-Minsky-style distance formula, thereby obtaining a new proof of the distance formula in the mapping class group without relying on the Masur-Minsky hiera"},"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":"1509.00632","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2015-09-02T10:34:37Z","cross_cats_sorted":[],"title_canon_sha256":"7b8ab67248a2c0b9b05d2a3ebc181b7bdea8c706077613629377b2165b8acb7b","abstract_canon_sha256":"58eb39ba3b98c310371650718499caeb5f67a9c93100d7dbe9734c0c57fc114e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:44:15.781489Z","signature_b64":"KykrsbmjV/e1XY3h1FCt0dPmeiWNoQ/M7Hfk08sXHNCJ1MxxoZDlfn/A4XuLgmlL9pMjA81GmSv/48/aFinzDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b30ecaa15a19816fff823187c1f2c4af88bc76d2dd54bd9be84f00c826a8042a","last_reissued_at":"2026-05-17T23:44:15.780874Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:44:15.780874Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Hierarchically hyperbolic spaces II: Combination theorems and the distance formula","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Alessandro Sisto, Jason Behrstock, Mark F. Hagen","submitted_at":"2015-09-02T10:34:37Z","abstract_excerpt":"We introduce a number of tools for finding and studying \\emph{hierarchically hyperbolic spaces (HHS)}, a rich class of spaces including mapping class groups of surfaces, Teichm\\\"{u}ller space with either the Teichm\\\"{u}ller or Weil-Petersson metrics, right-angled Artin groups, and the universal cover of any compact special cube complex. We begin by introducing a streamlined set of axioms defining an HHS. We prove that all HHSs satisfy a Masur-Minsky-style distance formula, thereby obtaining a new proof of the distance formula in the mapping class group without relying on the Masur-Minsky hiera"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.00632","kind":"arxiv","version":4},"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":"1509.00632","created_at":"2026-05-17T23:44:15.780960+00:00"},{"alias_kind":"arxiv_version","alias_value":"1509.00632v4","created_at":"2026-05-17T23:44:15.780960+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1509.00632","created_at":"2026-05-17T23:44:15.780960+00:00"},{"alias_kind":"pith_short_12","alias_value":"WMHMVIK2DGAW","created_at":"2026-05-18T12:29:47.479230+00:00"},{"alias_kind":"pith_short_16","alias_value":"WMHMVIK2DGAW774C","created_at":"2026-05-18T12:29:47.479230+00:00"},{"alias_kind":"pith_short_8","alias_value":"WMHMVIK2","created_at":"2026-05-18T12:29:47.479230+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/WMHMVIK2DGAW774CGGD4D4WEV6","json":"https://pith.science/pith/WMHMVIK2DGAW774CGGD4D4WEV6.json","graph_json":"https://pith.science/api/pith-number/WMHMVIK2DGAW774CGGD4D4WEV6/graph.json","events_json":"https://pith.science/api/pith-number/WMHMVIK2DGAW774CGGD4D4WEV6/events.json","paper":"https://pith.science/paper/WMHMVIK2"},"agent_actions":{"view_html":"https://pith.science/pith/WMHMVIK2DGAW774CGGD4D4WEV6","download_json":"https://pith.science/pith/WMHMVIK2DGAW774CGGD4D4WEV6.json","view_paper":"https://pith.science/paper/WMHMVIK2","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1509.00632&json=true","fetch_graph":"https://pith.science/api/pith-number/WMHMVIK2DGAW774CGGD4D4WEV6/graph.json","fetch_events":"https://pith.science/api/pith-number/WMHMVIK2DGAW774CGGD4D4WEV6/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WMHMVIK2DGAW774CGGD4D4WEV6/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WMHMVIK2DGAW774CGGD4D4WEV6/action/storage_attestation","attest_author":"https://pith.science/pith/WMHMVIK2DGAW774CGGD4D4WEV6/action/author_attestation","sign_citation":"https://pith.science/pith/WMHMVIK2DGAW774CGGD4D4WEV6/action/citation_signature","submit_replication":"https://pith.science/pith/WMHMVIK2DGAW774CGGD4D4WEV6/action/replication_record"}},"created_at":"2026-05-17T23:44:15.780960+00:00","updated_at":"2026-05-17T23:44:15.780960+00:00"}