{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2006:JKDN334JCK5TXUNIVAOG6A5WKX","short_pith_number":"pith:JKDN334J","canonical_record":{"source":{"id":"math/0607528","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.GR","submitted_at":"2006-07-21T13:41:39Z","cross_cats_sorted":[],"title_canon_sha256":"cc290d20c071ae2af099d4c7270154de939a7a92e7fb44dd7f664963a97bf27d","abstract_canon_sha256":"d6202a9f910229df05a277331a34a8fb9c9f851af60658b2fc07964b176a4ebe"},"schema_version":"1.0"},"canonical_sha256":"4a86ddef8912bb3bd1a8a81c6f03b655d849d7a4770d1031fb14b6c47bf90a72","source":{"kind":"arxiv","id":"math/0607528","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0607528","created_at":"2026-05-18T02:57:45Z"},{"alias_kind":"arxiv_version","alias_value":"math/0607528v2","created_at":"2026-05-18T02:57:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0607528","created_at":"2026-05-18T02:57:45Z"},{"alias_kind":"pith_short_12","alias_value":"JKDN334JCK5T","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_16","alias_value":"JKDN334JCK5TXUNI","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_8","alias_value":"JKDN334J","created_at":"2026-05-18T12:25:54Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2006:JKDN334JCK5TXUNIVAOG6A5WKX","target":"record","payload":{"canonical_record":{"source":{"id":"math/0607528","kind":"arxiv","version":2},"metadata":{"license":"","primary_cat":"math.GR","submitted_at":"2006-07-21T13:41:39Z","cross_cats_sorted":[],"title_canon_sha256":"cc290d20c071ae2af099d4c7270154de939a7a92e7fb44dd7f664963a97bf27d","abstract_canon_sha256":"d6202a9f910229df05a277331a34a8fb9c9f851af60658b2fc07964b176a4ebe"},"schema_version":"1.0"},"canonical_sha256":"4a86ddef8912bb3bd1a8a81c6f03b655d849d7a4770d1031fb14b6c47bf90a72","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:57:45.559805Z","signature_b64":"PRiBzFFYQdI32MqlVOMQQ9eLHacxyCbrwURl85bX7d+NWq9bj3ecsSaF2x2zOU5MIqxdkvYgcPAy/sFffFQ2Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4a86ddef8912bb3bd1a8a81c6f03b655d849d7a4770d1031fb14b6c47bf90a72","last_reissued_at":"2026-05-18T02:57:45.559232Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:57:45.559232Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"math/0607528","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":"2026-05-18T02:57:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LhRkzMDOC3yL8GUQfsLhL+b9hpRnNDAu7AWJNM9lq779uEamqY0WxUyCuEF2hHSwXBP6R03ZHZgDpGauw15pAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T20:48:28.977682Z"},"content_sha256":"126dabe40eadfb82ab173f6a6ac41fcdcd9ef05df011089c0a705e938ff5dbdc","schema_version":"1.0","event_id":"sha256:126dabe40eadfb82ab173f6a6ac41fcdcd9ef05df011089c0a705e938ff5dbdc"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2006:JKDN334JCK5TXUNIVAOG6A5WKX","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Tits alternative for non-spherical Pride groups","license":"","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Gerald Williams, Natalia Kopteva","submitted_at":"2006-07-21T13:41:39Z","abstract_excerpt":"Pride groups, or ``groups given by presentations in which each defining relator involves at most two types of generators'', include Coxeter groups, Artin groups, triangles of groups, and Vinberg's groups defined by periodic paired relations. We show that every non-spherical Pride group that is not a triangle of groups satisfies the Tits alternative."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0607528","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:57:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kyz5EOEoUMh4KPRZ89DUXZG6oS9eKABG1y6b9D0TD+IvlMsP6tK4VItG989uUURt/qSGQDBVCp9gUZtQ3iTjCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T20:48:28.978336Z"},"content_sha256":"3772aa1f1737585075a5ec241f0a84460e4316364be781984705ac9e7cc7e896","schema_version":"1.0","event_id":"sha256:3772aa1f1737585075a5ec241f0a84460e4316364be781984705ac9e7cc7e896"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JKDN334JCK5TXUNIVAOG6A5WKX/bundle.json","state_url":"https://pith.science/pith/JKDN334JCK5TXUNIVAOG6A5WKX/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JKDN334JCK5TXUNIVAOG6A5WKX/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":"2026-05-28T20:48:28Z","links":{"resolver":"https://pith.science/pith/JKDN334JCK5TXUNIVAOG6A5WKX","bundle":"https://pith.science/pith/JKDN334JCK5TXUNIVAOG6A5WKX/bundle.json","state":"https://pith.science/pith/JKDN334JCK5TXUNIVAOG6A5WKX/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JKDN334JCK5TXUNIVAOG6A5WKX/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2006:JKDN334JCK5TXUNIVAOG6A5WKX","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":"d6202a9f910229df05a277331a34a8fb9c9f851af60658b2fc07964b176a4ebe","cross_cats_sorted":[],"license":"","primary_cat":"math.GR","submitted_at":"2006-07-21T13:41:39Z","title_canon_sha256":"cc290d20c071ae2af099d4c7270154de939a7a92e7fb44dd7f664963a97bf27d"},"schema_version":"1.0","source":{"id":"math/0607528","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0607528","created_at":"2026-05-18T02:57:45Z"},{"alias_kind":"arxiv_version","alias_value":"math/0607528v2","created_at":"2026-05-18T02:57:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0607528","created_at":"2026-05-18T02:57:45Z"},{"alias_kind":"pith_short_12","alias_value":"JKDN334JCK5T","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_16","alias_value":"JKDN334JCK5TXUNI","created_at":"2026-05-18T12:25:54Z"},{"alias_kind":"pith_short_8","alias_value":"JKDN334J","created_at":"2026-05-18T12:25:54Z"}],"graph_snapshots":[{"event_id":"sha256:3772aa1f1737585075a5ec241f0a84460e4316364be781984705ac9e7cc7e896","target":"graph","created_at":"2026-05-18T02:57:45Z","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":"Pride groups, or ``groups given by presentations in which each defining relator involves at most two types of generators'', include Coxeter groups, Artin groups, triangles of groups, and Vinberg's groups defined by periodic paired relations. We show that every non-spherical Pride group that is not a triangle of groups satisfies the Tits alternative.","authors_text":"Gerald Williams, Natalia Kopteva","cross_cats":[],"headline":"","license":"","primary_cat":"math.GR","submitted_at":"2006-07-21T13:41:39Z","title":"The Tits alternative for non-spherical Pride groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0607528","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":"sha256:126dabe40eadfb82ab173f6a6ac41fcdcd9ef05df011089c0a705e938ff5dbdc","target":"record","created_at":"2026-05-18T02:57:45Z","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":"d6202a9f910229df05a277331a34a8fb9c9f851af60658b2fc07964b176a4ebe","cross_cats_sorted":[],"license":"","primary_cat":"math.GR","submitted_at":"2006-07-21T13:41:39Z","title_canon_sha256":"cc290d20c071ae2af099d4c7270154de939a7a92e7fb44dd7f664963a97bf27d"},"schema_version":"1.0","source":{"id":"math/0607528","kind":"arxiv","version":2}},"canonical_sha256":"4a86ddef8912bb3bd1a8a81c6f03b655d849d7a4770d1031fb14b6c47bf90a72","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4a86ddef8912bb3bd1a8a81c6f03b655d849d7a4770d1031fb14b6c47bf90a72","first_computed_at":"2026-05-18T02:57:45.559232Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:57:45.559232Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"PRiBzFFYQdI32MqlVOMQQ9eLHacxyCbrwURl85bX7d+NWq9bj3ecsSaF2x2zOU5MIqxdkvYgcPAy/sFffFQ2Bw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:57:45.559805Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0607528","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:126dabe40eadfb82ab173f6a6ac41fcdcd9ef05df011089c0a705e938ff5dbdc","sha256:3772aa1f1737585075a5ec241f0a84460e4316364be781984705ac9e7cc7e896"],"state_sha256":"234f5d65ce3e6a98bccff4ee37cf451c2eb84b139dbd1d752a925fa81c3371f8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2KILDewWF5C9cWXXeyMQ9ihTwTYQENn0VxLpvLD8ycQ98c/HRJyTvDtJpb1NMANOaseqHdDl3r94y6t/My1WBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T20:48:28.981630Z","bundle_sha256":"285d9f156df2765d6cf166fda5a1f3d4c37569656602298fc04d6b5cc5caadcb"}}