{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:PCWDOV7Y2U2UMKNZJDA6I5CK2E","short_pith_number":"pith:PCWDOV7Y","canonical_record":{"source":{"id":"1610.06245","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-10-19T23:42:31Z","cross_cats_sorted":["math.GT"],"title_canon_sha256":"e67720c44bd64f0b8b0ece9d1744c8c8b409660be5e266964606fb54f0c0e695","abstract_canon_sha256":"be7ffa80e6711f489ae6be7896a557d4aae63a062f754eb8f9ebcf72777b0d64"},"schema_version":"1.0"},"canonical_sha256":"78ac3757f8d5354629b948c1e4744ad12416b7c8d362681bfde4ec3a6345477c","source":{"kind":"arxiv","id":"1610.06245","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1610.06245","created_at":"2026-05-18T00:33:39Z"},{"alias_kind":"arxiv_version","alias_value":"1610.06245v2","created_at":"2026-05-18T00:33:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.06245","created_at":"2026-05-18T00:33:39Z"},{"alias_kind":"pith_short_12","alias_value":"PCWDOV7Y2U2U","created_at":"2026-05-18T12:30:39Z"},{"alias_kind":"pith_short_16","alias_value":"PCWDOV7Y2U2UMKNZ","created_at":"2026-05-18T12:30:39Z"},{"alias_kind":"pith_short_8","alias_value":"PCWDOV7Y","created_at":"2026-05-18T12:30:39Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:PCWDOV7Y2U2UMKNZJDA6I5CK2E","target":"record","payload":{"canonical_record":{"source":{"id":"1610.06245","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-10-19T23:42:31Z","cross_cats_sorted":["math.GT"],"title_canon_sha256":"e67720c44bd64f0b8b0ece9d1744c8c8b409660be5e266964606fb54f0c0e695","abstract_canon_sha256":"be7ffa80e6711f489ae6be7896a557d4aae63a062f754eb8f9ebcf72777b0d64"},"schema_version":"1.0"},"canonical_sha256":"78ac3757f8d5354629b948c1e4744ad12416b7c8d362681bfde4ec3a6345477c","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:33:39.583979Z","signature_b64":"djaxRXU+CMTp3yBImakhuTrtuuAxXExRF9eUojkYEQogMusQ+4bJA8t3iverkaMNfBjE5J8ioISZShsEN1DvBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"78ac3757f8d5354629b948c1e4744ad12416b7c8d362681bfde4ec3a6345477c","last_reissued_at":"2026-05-18T00:33:39.583368Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:33:39.583368Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1610.06245","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":"2026-05-18T00:33:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Wu2uvQXS1f41v8WdNOoOJDrcCoTI4DLyVeQfq0lu00c6CtvUgaKlaKc5ib6pKQkQ6s2r/0oAnlaS6l1pc4QDDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T21:38:07.384152Z"},"content_sha256":"1529b5733f84f665e1410518c7a367649a6fd5488204c96d8b14fb3e2a4e9588","schema_version":"1.0","event_id":"sha256:1529b5733f84f665e1410518c7a367649a6fd5488204c96d8b14fb3e2a4e9588"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:PCWDOV7Y2U2UMKNZJDA6I5CK2E","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Commensurability for certain right-angled Coxeter groups and geometric amalgams of free groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GT"],"primary_cat":"math.GR","authors_text":"Anne Thomas, Emily Stark, Pallavi Dani","submitted_at":"2016-10-19T23:42:31Z","abstract_excerpt":"We give explicit necessary and sufficient conditions for the abstract commensurability of certain families of 1-ended, hyperbolic groups, namely right-angled Coxeter groups defined by generalized theta-graphs and cycles of generalized theta-graphs, and geometric amalgams of free groups whose JSJ graphs are trees of diameter at most 4. We also show that if a geometric amalgam of free groups has JSJ graph a tree, then it is commensurable to a right-angled Coxeter group, and give an example of a geometric amalgam of free groups which is not quasi-isometric (hence not commensurable) to any group w"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.06245","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":"2026-05-18T00:33:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4jZf2Z2RM59T7xo4Fu2CrTpeB2csSB5WUwmEilm+nVH2LCOvWwsSWWjEvIVmeWmwipUoT/VJE+v5CQrLGFm8BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T21:38:07.384829Z"},"content_sha256":"09d11ddc2ec592828dc99f59f0b4f032d929bae4842ce7a44a073ab9401eab83","schema_version":"1.0","event_id":"sha256:09d11ddc2ec592828dc99f59f0b4f032d929bae4842ce7a44a073ab9401eab83"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PCWDOV7Y2U2UMKNZJDA6I5CK2E/bundle.json","state_url":"https://pith.science/pith/PCWDOV7Y2U2UMKNZJDA6I5CK2E/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PCWDOV7Y2U2UMKNZJDA6I5CK2E/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":"2026-05-30T21:38:07Z","links":{"resolver":"https://pith.science/pith/PCWDOV7Y2U2UMKNZJDA6I5CK2E","bundle":"https://pith.science/pith/PCWDOV7Y2U2UMKNZJDA6I5CK2E/bundle.json","state":"https://pith.science/pith/PCWDOV7Y2U2UMKNZJDA6I5CK2E/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PCWDOV7Y2U2UMKNZJDA6I5CK2E/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:PCWDOV7Y2U2UMKNZJDA6I5CK2E","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":"be7ffa80e6711f489ae6be7896a557d4aae63a062f754eb8f9ebcf72777b0d64","cross_cats_sorted":["math.GT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-10-19T23:42:31Z","title_canon_sha256":"e67720c44bd64f0b8b0ece9d1744c8c8b409660be5e266964606fb54f0c0e695"},"schema_version":"1.0","source":{"id":"1610.06245","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1610.06245","created_at":"2026-05-18T00:33:39Z"},{"alias_kind":"arxiv_version","alias_value":"1610.06245v2","created_at":"2026-05-18T00:33:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.06245","created_at":"2026-05-18T00:33:39Z"},{"alias_kind":"pith_short_12","alias_value":"PCWDOV7Y2U2U","created_at":"2026-05-18T12:30:39Z"},{"alias_kind":"pith_short_16","alias_value":"PCWDOV7Y2U2UMKNZ","created_at":"2026-05-18T12:30:39Z"},{"alias_kind":"pith_short_8","alias_value":"PCWDOV7Y","created_at":"2026-05-18T12:30:39Z"}],"graph_snapshots":[{"event_id":"sha256:09d11ddc2ec592828dc99f59f0b4f032d929bae4842ce7a44a073ab9401eab83","target":"graph","created_at":"2026-05-18T00:33:39Z","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 give explicit necessary and sufficient conditions for the abstract commensurability of certain families of 1-ended, hyperbolic groups, namely right-angled Coxeter groups defined by generalized theta-graphs and cycles of generalized theta-graphs, and geometric amalgams of free groups whose JSJ graphs are trees of diameter at most 4. We also show that if a geometric amalgam of free groups has JSJ graph a tree, then it is commensurable to a right-angled Coxeter group, and give an example of a geometric amalgam of free groups which is not quasi-isometric (hence not commensurable) to any group w","authors_text":"Anne Thomas, Emily Stark, Pallavi Dani","cross_cats":["math.GT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-10-19T23:42:31Z","title":"Commensurability for certain right-angled Coxeter groups and geometric amalgams of free groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.06245","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":"sha256:1529b5733f84f665e1410518c7a367649a6fd5488204c96d8b14fb3e2a4e9588","target":"record","created_at":"2026-05-18T00:33:39Z","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":"be7ffa80e6711f489ae6be7896a557d4aae63a062f754eb8f9ebcf72777b0d64","cross_cats_sorted":["math.GT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-10-19T23:42:31Z","title_canon_sha256":"e67720c44bd64f0b8b0ece9d1744c8c8b409660be5e266964606fb54f0c0e695"},"schema_version":"1.0","source":{"id":"1610.06245","kind":"arxiv","version":2}},"canonical_sha256":"78ac3757f8d5354629b948c1e4744ad12416b7c8d362681bfde4ec3a6345477c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"78ac3757f8d5354629b948c1e4744ad12416b7c8d362681bfde4ec3a6345477c","first_computed_at":"2026-05-18T00:33:39.583368Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:33:39.583368Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"djaxRXU+CMTp3yBImakhuTrtuuAxXExRF9eUojkYEQogMusQ+4bJA8t3iverkaMNfBjE5J8ioISZShsEN1DvBA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:33:39.583979Z","signed_message":"canonical_sha256_bytes"},"source_id":"1610.06245","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1529b5733f84f665e1410518c7a367649a6fd5488204c96d8b14fb3e2a4e9588","sha256:09d11ddc2ec592828dc99f59f0b4f032d929bae4842ce7a44a073ab9401eab83"],"state_sha256":"e1ab49296f658ae955f063fd0cc90bf0fe12993aa7ab0b66c397a6ea4846c908"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YuHcnap6RoKMBLJqIoD03ac64rF7HnQYufng20YDLkrtUU/dYKsazqKmSWz0k1Q/t9JYqEVFImra1/rDS+THBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T21:38:07.390124Z","bundle_sha256":"bf910744541ca1b9323b4344d8eac04f5e0609fef1247bd23506793ecca91730"}}