{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:CX6QI2B5UOXCHKAZ7LSSTX5AE2","short_pith_number":"pith:CX6QI2B5","canonical_record":{"source":{"id":"1810.06476","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2018-10-15T15:47:36Z","cross_cats_sorted":[],"title_canon_sha256":"e4f28d93b94e5916bc5a281cb879524a2007e23925a51394d922cc28cb87225c","abstract_canon_sha256":"d01f11c69e079a7b20e47ea28343c2e4d76ddf616505f61c34e487ad7c5d1fad"},"schema_version":"1.0"},"canonical_sha256":"15fd04683da3ae23a819fae529dfa02687b464f1c49bc672e1a5e09157a64628","source":{"kind":"arxiv","id":"1810.06476","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1810.06476","created_at":"2026-07-05T00:06:31Z"},{"alias_kind":"arxiv_version","alias_value":"1810.06476v2","created_at":"2026-07-05T00:06:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1810.06476","created_at":"2026-07-05T00:06:31Z"},{"alias_kind":"pith_short_12","alias_value":"CX6QI2B5UOXC","created_at":"2026-07-05T00:06:31Z"},{"alias_kind":"pith_short_16","alias_value":"CX6QI2B5UOXCHKAZ","created_at":"2026-07-05T00:06:31Z"},{"alias_kind":"pith_short_8","alias_value":"CX6QI2B5","created_at":"2026-07-05T00:06:31Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:CX6QI2B5UOXCHKAZ7LSSTX5AE2","target":"record","payload":{"canonical_record":{"source":{"id":"1810.06476","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2018-10-15T15:47:36Z","cross_cats_sorted":[],"title_canon_sha256":"e4f28d93b94e5916bc5a281cb879524a2007e23925a51394d922cc28cb87225c","abstract_canon_sha256":"d01f11c69e079a7b20e47ea28343c2e4d76ddf616505f61c34e487ad7c5d1fad"},"schema_version":"1.0"},"canonical_sha256":"15fd04683da3ae23a819fae529dfa02687b464f1c49bc672e1a5e09157a64628","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T00:06:31.415593Z","signature_b64":"ItK7Ez18IqRvrtMcWvQ/4e3DqnnJ+ASeEe+KoLeidrROphAAtH1fieJvMDMhgOjoCfnOBTJbmF204FyGcKI5CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"15fd04683da3ae23a819fae529dfa02687b464f1c49bc672e1a5e09157a64628","last_reissued_at":"2026-07-05T00:06:31.415147Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T00:06:31.415147Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1810.06476","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T00:06:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"w4GQcNXxnyDS+F1fhJ79ItYCou04e1+x5pwWpIOYR9EXc0HUvLGvSV8Cb7uYUGkC2LZiOtCTDNTgMXNdUnLsCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-18T11:53:13.359012Z"},"content_sha256":"0ce50b00c2b94e35151e9fd53e389c22b52629c26ca17eefee3ea73f3495a4a9","schema_version":"1.0","event_id":"sha256:0ce50b00c2b94e35151e9fd53e389c22b52629c26ca17eefee3ea73f3495a4a9"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:CX6QI2B5UOXCHKAZ7LSSTX5AE2","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A refined combination theorem for hierarchically hyperbolic groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Bruno Robbio, Federico Berlai","submitted_at":"2018-10-15T15:47:36Z","abstract_excerpt":"In this work, we are concerned with hierarchically hyperbolic spaces and hierarchically hyperbolic groups. Our main result is a wide generalization of a combination theorem of Behrstock, Hagen, and Sisto. In particular, as a consequence, we show that any finite graph product of hierarchically hyperbolic groups is again a hierarchically hyperbolic group, thereby answering a question posed by Behrstock, Hagen, and Sisto. In order to operate in such a general setting, we establish a number of structural results for hierarchically hyperbolic spaces and hieromorphisms (that is, morphisms between su"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.06476","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1810.06476/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T00:06:31Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BTgLUS4YfIfyyFTGRSHS6lUEMAcl/tBQXq1lMGEhISTYjkq3Ut9jEumrrsvrl3umjfRsRuZd5QWY/nDvL+fHBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-18T11:53:13.359925Z"},"content_sha256":"c20b3526ac0383b347684760ce997f7960b6f56785b77a54a19bf69d6d611e9c","schema_version":"1.0","event_id":"sha256:c20b3526ac0383b347684760ce997f7960b6f56785b77a54a19bf69d6d611e9c"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CX6QI2B5UOXCHKAZ7LSSTX5AE2/bundle.json","state_url":"https://pith.science/pith/CX6QI2B5UOXCHKAZ7LSSTX5AE2/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CX6QI2B5UOXCHKAZ7LSSTX5AE2/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-08-18T11:53:13Z","links":{"resolver":"https://pith.science/pith/CX6QI2B5UOXCHKAZ7LSSTX5AE2","bundle":"https://pith.science/pith/CX6QI2B5UOXCHKAZ7LSSTX5AE2/bundle.json","state":"https://pith.science/pith/CX6QI2B5UOXCHKAZ7LSSTX5AE2/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CX6QI2B5UOXCHKAZ7LSSTX5AE2/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:CX6QI2B5UOXCHKAZ7LSSTX5AE2","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"d01f11c69e079a7b20e47ea28343c2e4d76ddf616505f61c34e487ad7c5d1fad","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2018-10-15T15:47:36Z","title_canon_sha256":"e4f28d93b94e5916bc5a281cb879524a2007e23925a51394d922cc28cb87225c"},"schema_version":"1.0","source":{"id":"1810.06476","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1810.06476","created_at":"2026-07-05T00:06:31Z"},{"alias_kind":"arxiv_version","alias_value":"1810.06476v2","created_at":"2026-07-05T00:06:31Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1810.06476","created_at":"2026-07-05T00:06:31Z"},{"alias_kind":"pith_short_12","alias_value":"CX6QI2B5UOXC","created_at":"2026-07-05T00:06:31Z"},{"alias_kind":"pith_short_16","alias_value":"CX6QI2B5UOXCHKAZ","created_at":"2026-07-05T00:06:31Z"},{"alias_kind":"pith_short_8","alias_value":"CX6QI2B5","created_at":"2026-07-05T00:06:31Z"}],"graph_snapshots":[{"event_id":"sha256:c20b3526ac0383b347684760ce997f7960b6f56785b77a54a19bf69d6d611e9c","target":"graph","created_at":"2026-07-05T00:06:31Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/1810.06476/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this work, we are concerned with hierarchically hyperbolic spaces and hierarchically hyperbolic groups. Our main result is a wide generalization of a combination theorem of Behrstock, Hagen, and Sisto. In particular, as a consequence, we show that any finite graph product of hierarchically hyperbolic groups is again a hierarchically hyperbolic group, thereby answering a question posed by Behrstock, Hagen, and Sisto. In order to operate in such a general setting, we establish a number of structural results for hierarchically hyperbolic spaces and hieromorphisms (that is, morphisms between su","authors_text":"Bruno Robbio, Federico Berlai","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2018-10-15T15:47:36Z","title":"A refined combination theorem for hierarchically hyperbolic groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.06476","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:0ce50b00c2b94e35151e9fd53e389c22b52629c26ca17eefee3ea73f3495a4a9","target":"record","created_at":"2026-07-05T00:06:31Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"d01f11c69e079a7b20e47ea28343c2e4d76ddf616505f61c34e487ad7c5d1fad","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2018-10-15T15:47:36Z","title_canon_sha256":"e4f28d93b94e5916bc5a281cb879524a2007e23925a51394d922cc28cb87225c"},"schema_version":"1.0","source":{"id":"1810.06476","kind":"arxiv","version":2}},"canonical_sha256":"15fd04683da3ae23a819fae529dfa02687b464f1c49bc672e1a5e09157a64628","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"15fd04683da3ae23a819fae529dfa02687b464f1c49bc672e1a5e09157a64628","first_computed_at":"2026-07-05T00:06:31.415147Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T00:06:31.415147Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ItK7Ez18IqRvrtMcWvQ/4e3DqnnJ+ASeEe+KoLeidrROphAAtH1fieJvMDMhgOjoCfnOBTJbmF204FyGcKI5CQ==","signature_status":"signed_v1","signed_at":"2026-07-05T00:06:31.415593Z","signed_message":"canonical_sha256_bytes"},"source_id":"1810.06476","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0ce50b00c2b94e35151e9fd53e389c22b52629c26ca17eefee3ea73f3495a4a9","sha256:c20b3526ac0383b347684760ce997f7960b6f56785b77a54a19bf69d6d611e9c"],"state_sha256":"82ae7ae1974aaf2de623464bd3f5686c848106c9c7873408832b5e655e018853"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/m/1Vg+LAFCD7EBUodqOQVhCVLyjR3zM34ufjFXPObaFVk2pIbtqKjY0eiNDcryn5lky93qA7+KWsNZDcqzTAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-18T11:53:13.368749Z","bundle_sha256":"80ced7cae44a5ba3567d5b7d22c9dfeb975d7ac4f460b8093235d7f60ce09d31"}}