{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:DK5ZWEETRHX2ANZ7WYNQCW2ADX","short_pith_number":"pith:DK5ZWEET","canonical_record":{"source":{"id":"1002.3991","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-02-21T18:57:24Z","cross_cats_sorted":[],"title_canon_sha256":"f05e8be1ceaefe9c9dee7245a0a73d87a8ccb0d55a9fbbec0979344e577612f4","abstract_canon_sha256":"86ea7c520f6768ec40d76294c82946a6aea20245bdb75e66b7b6b48427f54cce"},"schema_version":"1.0"},"canonical_sha256":"1abb9b109389efa0373fb61b015b401df2e3247408d48a9a5596be02f6768950","source":{"kind":"arxiv","id":"1002.3991","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1002.3991","created_at":"2026-05-18T04:00:49Z"},{"alias_kind":"arxiv_version","alias_value":"1002.3991v1","created_at":"2026-05-18T04:00:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1002.3991","created_at":"2026-05-18T04:00:49Z"},{"alias_kind":"pith_short_12","alias_value":"DK5ZWEETRHX2","created_at":"2026-05-18T12:26:06Z"},{"alias_kind":"pith_short_16","alias_value":"DK5ZWEETRHX2ANZ7","created_at":"2026-05-18T12:26:06Z"},{"alias_kind":"pith_short_8","alias_value":"DK5ZWEET","created_at":"2026-05-18T12:26:06Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:DK5ZWEETRHX2ANZ7WYNQCW2ADX","target":"record","payload":{"canonical_record":{"source":{"id":"1002.3991","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-02-21T18:57:24Z","cross_cats_sorted":[],"title_canon_sha256":"f05e8be1ceaefe9c9dee7245a0a73d87a8ccb0d55a9fbbec0979344e577612f4","abstract_canon_sha256":"86ea7c520f6768ec40d76294c82946a6aea20245bdb75e66b7b6b48427f54cce"},"schema_version":"1.0"},"canonical_sha256":"1abb9b109389efa0373fb61b015b401df2e3247408d48a9a5596be02f6768950","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:00:49.672470Z","signature_b64":"8UGdxX9bzNvGlz3yd3TGF09dAu7ZyKmJn4m/DwySKZZl42CFo+NlCt/dC6m4czbWLq9AQFUxzy4cgrFDcqE2BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1abb9b109389efa0373fb61b015b401df2e3247408d48a9a5596be02f6768950","last_reissued_at":"2026-05-18T04:00:49.671761Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:00:49.671761Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1002.3991","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:00:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8ODD8+qsKMNvQrldgE5oXHxQzznbVG59m1W42Ywber88T1prKb3AYX6kEl/OWq+mvL5cFjOrZAemtZkVj1oaAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T15:00:17.986441Z"},"content_sha256":"a6d8919c359396ebee9fb9d9e72b9f94fbb8c3eb87126f51be9183e3eca20761","schema_version":"1.0","event_id":"sha256:a6d8919c359396ebee9fb9d9e72b9f94fbb8c3eb87126f51be9183e3eca20761"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:DK5ZWEETRHX2ANZ7WYNQCW2ADX","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Bipolar Coxeter groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Pierre-Emmanuel Caprace, Piotr Przytycki","submitted_at":"2010-02-21T18:57:24Z","abstract_excerpt":"We consider the class of those Coxeter groups for which removing from the Cayley graph any tubular neighbourhood of any wall leaves exactly two connected components. We call these Coxeter groups bipolar. They include both the virtually Poincare duality Coxeter groups and the infinite irreducible 2-spherical ones. We show in a geometric way that a bipolar Coxeter group admits a unique conjugacy class of Coxeter generating sets. Moreover, we provide a characterisation of bipolar Coxeter groups in terms of the associated Coxeter diagram."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1002.3991","kind":"arxiv","version":1},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:00:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"OsF3BVGEhjOr5pS5pXaH4MHFO6cHVwBDyuIxOtiAzqSx4lqumKNxPWYzC2C3JvQwn0vZxEze7W576kk9NDxCAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T15:00:17.986861Z"},"content_sha256":"533d5ee31b270cc0a8354d041de91fea5c5761e4506199ca83273152cfac2046","schema_version":"1.0","event_id":"sha256:533d5ee31b270cc0a8354d041de91fea5c5761e4506199ca83273152cfac2046"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/DK5ZWEETRHX2ANZ7WYNQCW2ADX/bundle.json","state_url":"https://pith.science/pith/DK5ZWEETRHX2ANZ7WYNQCW2ADX/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/DK5ZWEETRHX2ANZ7WYNQCW2ADX/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":"2026-05-27T15:00:17Z","links":{"resolver":"https://pith.science/pith/DK5ZWEETRHX2ANZ7WYNQCW2ADX","bundle":"https://pith.science/pith/DK5ZWEETRHX2ANZ7WYNQCW2ADX/bundle.json","state":"https://pith.science/pith/DK5ZWEETRHX2ANZ7WYNQCW2ADX/state.json","well_known_bundle":"https://pith.science/.well-known/pith/DK5ZWEETRHX2ANZ7WYNQCW2ADX/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:DK5ZWEETRHX2ANZ7WYNQCW2ADX","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":"86ea7c520f6768ec40d76294c82946a6aea20245bdb75e66b7b6b48427f54cce","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-02-21T18:57:24Z","title_canon_sha256":"f05e8be1ceaefe9c9dee7245a0a73d87a8ccb0d55a9fbbec0979344e577612f4"},"schema_version":"1.0","source":{"id":"1002.3991","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1002.3991","created_at":"2026-05-18T04:00:49Z"},{"alias_kind":"arxiv_version","alias_value":"1002.3991v1","created_at":"2026-05-18T04:00:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1002.3991","created_at":"2026-05-18T04:00:49Z"},{"alias_kind":"pith_short_12","alias_value":"DK5ZWEETRHX2","created_at":"2026-05-18T12:26:06Z"},{"alias_kind":"pith_short_16","alias_value":"DK5ZWEETRHX2ANZ7","created_at":"2026-05-18T12:26:06Z"},{"alias_kind":"pith_short_8","alias_value":"DK5ZWEET","created_at":"2026-05-18T12:26:06Z"}],"graph_snapshots":[{"event_id":"sha256:533d5ee31b270cc0a8354d041de91fea5c5761e4506199ca83273152cfac2046","target":"graph","created_at":"2026-05-18T04:00:49Z","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"},"paper":{"abstract_excerpt":"We consider the class of those Coxeter groups for which removing from the Cayley graph any tubular neighbourhood of any wall leaves exactly two connected components. We call these Coxeter groups bipolar. They include both the virtually Poincare duality Coxeter groups and the infinite irreducible 2-spherical ones. We show in a geometric way that a bipolar Coxeter group admits a unique conjugacy class of Coxeter generating sets. Moreover, we provide a characterisation of bipolar Coxeter groups in terms of the associated Coxeter diagram.","authors_text":"Pierre-Emmanuel Caprace, Piotr Przytycki","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-02-21T18:57:24Z","title":"Bipolar Coxeter groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1002.3991","kind":"arxiv","version":1},"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:a6d8919c359396ebee9fb9d9e72b9f94fbb8c3eb87126f51be9183e3eca20761","target":"record","created_at":"2026-05-18T04:00:49Z","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":"86ea7c520f6768ec40d76294c82946a6aea20245bdb75e66b7b6b48427f54cce","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-02-21T18:57:24Z","title_canon_sha256":"f05e8be1ceaefe9c9dee7245a0a73d87a8ccb0d55a9fbbec0979344e577612f4"},"schema_version":"1.0","source":{"id":"1002.3991","kind":"arxiv","version":1}},"canonical_sha256":"1abb9b109389efa0373fb61b015b401df2e3247408d48a9a5596be02f6768950","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1abb9b109389efa0373fb61b015b401df2e3247408d48a9a5596be02f6768950","first_computed_at":"2026-05-18T04:00:49.671761Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:00:49.671761Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8UGdxX9bzNvGlz3yd3TGF09dAu7ZyKmJn4m/DwySKZZl42CFo+NlCt/dC6m4czbWLq9AQFUxzy4cgrFDcqE2BQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:00:49.672470Z","signed_message":"canonical_sha256_bytes"},"source_id":"1002.3991","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a6d8919c359396ebee9fb9d9e72b9f94fbb8c3eb87126f51be9183e3eca20761","sha256:533d5ee31b270cc0a8354d041de91fea5c5761e4506199ca83273152cfac2046"],"state_sha256":"03d03941fac2324d293ab52c367f796bde52085d4fec0b668ab92a9c5cc00cc5"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xxNRN/DFmuRurzfg9cDGp03buDUKgZToh5UBlTJe4oAfkEupquQdueWPsWVRi/B6M0GmvpLyrZr51ceHfHLmCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T15:00:17.989820Z","bundle_sha256":"e1f58e43aa89173a2ccbbfbdbca0ff966f898fc6269a6304c46e4b4763f69556"}}