{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:QNNUDZEQDTSXSOSFGCMOJ5IGS4","short_pith_number":"pith:QNNUDZEQ","canonical_record":{"source":{"id":"1212.2618","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2012-12-11T20:37:00Z","cross_cats_sorted":["math.DS","math.GT"],"title_canon_sha256":"d6761f571a0b34cb05b393ef2727ee37d1a633d9067311bc4938b26b6818991c","abstract_canon_sha256":"584cd6a6bddac77eaaf08ca8905f4dc59a7335a70ceb4854379bc29afeb0e152"},"schema_version":"1.0"},"canonical_sha256":"835b41e4901ce5793a453098e4f506970979fa7a23609995016f08f56842d3fb","source":{"kind":"arxiv","id":"1212.2618","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1212.2618","created_at":"2026-05-18T01:28:21Z"},{"alias_kind":"arxiv_version","alias_value":"1212.2618v2","created_at":"2026-05-18T01:28:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1212.2618","created_at":"2026-05-18T01:28:21Z"},{"alias_kind":"pith_short_12","alias_value":"QNNUDZEQDTSX","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_16","alias_value":"QNNUDZEQDTSXSOSF","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_8","alias_value":"QNNUDZEQ","created_at":"2026-05-18T12:27:18Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:QNNUDZEQDTSXSOSFGCMOJ5IGS4","target":"record","payload":{"canonical_record":{"source":{"id":"1212.2618","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2012-12-11T20:37:00Z","cross_cats_sorted":["math.DS","math.GT"],"title_canon_sha256":"d6761f571a0b34cb05b393ef2727ee37d1a633d9067311bc4938b26b6818991c","abstract_canon_sha256":"584cd6a6bddac77eaaf08ca8905f4dc59a7335a70ceb4854379bc29afeb0e152"},"schema_version":"1.0"},"canonical_sha256":"835b41e4901ce5793a453098e4f506970979fa7a23609995016f08f56842d3fb","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:28:21.404753Z","signature_b64":"gOamGGePeCO0PwKrgNR8Bw5Qm+CMF1Ofas+3rH9ivkvLxKAUlj9MhHVBHS3mwRLFfSxdyJV+vT/4iyrLSj1RBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"835b41e4901ce5793a453098e4f506970979fa7a23609995016f08f56842d3fb","last_reissued_at":"2026-05-18T01:28:21.404215Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:28:21.404215Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1212.2618","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":"2026-05-18T01:28:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HWelKF/RyssO2SHAR2mGGkQeSB0KiY4ldHctc6RJlMxGuj6xti+Z+UDUQKll8Rl1F6bGsmzdDfXapY2YZc46Ag==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T05:35:41.662576Z"},"content_sha256":"49ad967fc259c3f591e1d68e6276c4854c46e55234f4577bfe5f603592fc8416","schema_version":"1.0","event_id":"sha256:49ad967fc259c3f591e1d68e6276c4854c46e55234f4577bfe5f603592fc8416"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:QNNUDZEQDTSXSOSFGCMOJ5IGS4","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Surface subgroups from linear programming","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS","math.GT"],"primary_cat":"math.GR","authors_text":"Alden Walker, Danny Calegari","submitted_at":"2012-12-11T20:37:00Z","abstract_excerpt":"We show that certain classes of graphs of free groups contain surface subgroups, including groups with positive $b_2$ obtained by doubling free groups along collections of subgroups, and groups obtained by \"random\" ascending HNN extensions of free groups. A special case is the HNN extension associated to the endomorphism of a rank 2 free group sending a to ab and b to ba; this example (and the random examples) answer in the negative well-known questions of Sapir. We further show that the unit ball in the Gromov norm (in dimension 2) of a double of a free group along a collection of subgroups i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.2618","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":"2026-05-18T01:28:21Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SFrpC79o7s10rXiKvJ1Ai2+DW8pYNC7dIKSs2/ovuSPKlR+xB6WN1ndA2fPo4E5EaLufA4dxnGjLUtUAxx3zDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T05:35:41.663242Z"},"content_sha256":"d521e14585b5cc8a8ca54ccdb75fe40a1eba2e6de303dd1b0bcf5f89b845dc90","schema_version":"1.0","event_id":"sha256:d521e14585b5cc8a8ca54ccdb75fe40a1eba2e6de303dd1b0bcf5f89b845dc90"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QNNUDZEQDTSXSOSFGCMOJ5IGS4/bundle.json","state_url":"https://pith.science/pith/QNNUDZEQDTSXSOSFGCMOJ5IGS4/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QNNUDZEQDTSXSOSFGCMOJ5IGS4/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":"2026-05-27T05:35:41Z","links":{"resolver":"https://pith.science/pith/QNNUDZEQDTSXSOSFGCMOJ5IGS4","bundle":"https://pith.science/pith/QNNUDZEQDTSXSOSFGCMOJ5IGS4/bundle.json","state":"https://pith.science/pith/QNNUDZEQDTSXSOSFGCMOJ5IGS4/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QNNUDZEQDTSXSOSFGCMOJ5IGS4/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:QNNUDZEQDTSXSOSFGCMOJ5IGS4","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":"584cd6a6bddac77eaaf08ca8905f4dc59a7335a70ceb4854379bc29afeb0e152","cross_cats_sorted":["math.DS","math.GT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2012-12-11T20:37:00Z","title_canon_sha256":"d6761f571a0b34cb05b393ef2727ee37d1a633d9067311bc4938b26b6818991c"},"schema_version":"1.0","source":{"id":"1212.2618","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1212.2618","created_at":"2026-05-18T01:28:21Z"},{"alias_kind":"arxiv_version","alias_value":"1212.2618v2","created_at":"2026-05-18T01:28:21Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1212.2618","created_at":"2026-05-18T01:28:21Z"},{"alias_kind":"pith_short_12","alias_value":"QNNUDZEQDTSX","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_16","alias_value":"QNNUDZEQDTSXSOSF","created_at":"2026-05-18T12:27:18Z"},{"alias_kind":"pith_short_8","alias_value":"QNNUDZEQ","created_at":"2026-05-18T12:27:18Z"}],"graph_snapshots":[{"event_id":"sha256:d521e14585b5cc8a8ca54ccdb75fe40a1eba2e6de303dd1b0bcf5f89b845dc90","target":"graph","created_at":"2026-05-18T01:28:21Z","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 show that certain classes of graphs of free groups contain surface subgroups, including groups with positive $b_2$ obtained by doubling free groups along collections of subgroups, and groups obtained by \"random\" ascending HNN extensions of free groups. A special case is the HNN extension associated to the endomorphism of a rank 2 free group sending a to ab and b to ba; this example (and the random examples) answer in the negative well-known questions of Sapir. We further show that the unit ball in the Gromov norm (in dimension 2) of a double of a free group along a collection of subgroups i","authors_text":"Alden Walker, Danny Calegari","cross_cats":["math.DS","math.GT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2012-12-11T20:37:00Z","title":"Surface subgroups from linear programming"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.2618","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":"sha256:49ad967fc259c3f591e1d68e6276c4854c46e55234f4577bfe5f603592fc8416","target":"record","created_at":"2026-05-18T01:28:21Z","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":"584cd6a6bddac77eaaf08ca8905f4dc59a7335a70ceb4854379bc29afeb0e152","cross_cats_sorted":["math.DS","math.GT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2012-12-11T20:37:00Z","title_canon_sha256":"d6761f571a0b34cb05b393ef2727ee37d1a633d9067311bc4938b26b6818991c"},"schema_version":"1.0","source":{"id":"1212.2618","kind":"arxiv","version":2}},"canonical_sha256":"835b41e4901ce5793a453098e4f506970979fa7a23609995016f08f56842d3fb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"835b41e4901ce5793a453098e4f506970979fa7a23609995016f08f56842d3fb","first_computed_at":"2026-05-18T01:28:21.404215Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:28:21.404215Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"gOamGGePeCO0PwKrgNR8Bw5Qm+CMF1Ofas+3rH9ivkvLxKAUlj9MhHVBHS3mwRLFfSxdyJV+vT/4iyrLSj1RBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:28:21.404753Z","signed_message":"canonical_sha256_bytes"},"source_id":"1212.2618","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:49ad967fc259c3f591e1d68e6276c4854c46e55234f4577bfe5f603592fc8416","sha256:d521e14585b5cc8a8ca54ccdb75fe40a1eba2e6de303dd1b0bcf5f89b845dc90"],"state_sha256":"6d042689cc1a444235c820901cc4104a252a1a8fede19b5ecf214f231b365f06"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"e+iCcir10iSGuv+5vLFhgjcMkriNXHBff1Jzd52po72cwjdnc/7iVFbrgxMA0Mtmjzc18x1WlSjZsMjgmJ2yAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T05:35:41.666903Z","bundle_sha256":"214e1cb9b4d2469bf6de7cead3be92d551ac06fb0bec17b6308a1aec2d8832c0"}}