{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:XNNPBP6KK62KP34WE363OZ4TP6","short_pith_number":"pith:XNNPBP6K","canonical_record":{"source":{"id":"0903.1456","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2009-03-08T21:06:38Z","cross_cats_sorted":[],"title_canon_sha256":"8e5ead1acfcb798356c615091a9ea5060511fcbcad681b3988e0220766938441","abstract_canon_sha256":"8464e6ea3bc2954774c6367988c57701cd1adc436e52a6efb4232f02555df877"},"schema_version":"1.0"},"canonical_sha256":"bb5af0bfca57b4a7ef9626fdb767937fa020bbbb49990af92994841f6d794188","source":{"kind":"arxiv","id":"0903.1456","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0903.1456","created_at":"2026-05-18T03:33:27Z"},{"alias_kind":"arxiv_version","alias_value":"0903.1456v3","created_at":"2026-05-18T03:33:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0903.1456","created_at":"2026-05-18T03:33:27Z"},{"alias_kind":"pith_short_12","alias_value":"XNNPBP6KK62K","created_at":"2026-05-18T12:26:02Z"},{"alias_kind":"pith_short_16","alias_value":"XNNPBP6KK62KP34W","created_at":"2026-05-18T12:26:02Z"},{"alias_kind":"pith_short_8","alias_value":"XNNPBP6K","created_at":"2026-05-18T12:26:02Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:XNNPBP6KK62KP34WE363OZ4TP6","target":"record","payload":{"canonical_record":{"source":{"id":"0903.1456","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2009-03-08T21:06:38Z","cross_cats_sorted":[],"title_canon_sha256":"8e5ead1acfcb798356c615091a9ea5060511fcbcad681b3988e0220766938441","abstract_canon_sha256":"8464e6ea3bc2954774c6367988c57701cd1adc436e52a6efb4232f02555df877"},"schema_version":"1.0"},"canonical_sha256":"bb5af0bfca57b4a7ef9626fdb767937fa020bbbb49990af92994841f6d794188","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:33:27.020968Z","signature_b64":"g0EwEy3Edp7cVr5Ic0GQYSu7CFIQs1fE1b1F3tcNAbxgqZ8nGN3aBx1ISm0etW2rMgj2YzvOXZXut0+JGVgLCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bb5af0bfca57b4a7ef9626fdb767937fa020bbbb49990af92994841f6d794188","last_reissued_at":"2026-05-18T03:33:27.020183Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:33:27.020183Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0903.1456","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:33:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+66bxT0VPoJ9UdHYfJlU+fENncptRxlcRZ43P6Bh0z5opAoWOF8JtHzJz6JEb8CuUpnd5dt0auYdgeMzqNMkDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T17:41:16.711173Z"},"content_sha256":"a154b1fb144413666999757dd51c4a008e2edca714aa5407b6cdd71bd67a04d6","schema_version":"1.0","event_id":"sha256:a154b1fb144413666999757dd51c4a008e2edca714aa5407b6cdd71bd67a04d6"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:XNNPBP6KK62KP34WE363OZ4TP6","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On common fundamental domains","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Deguang Han, Dorin Ervin Dutkay, Gabriel Picioroaga, Palle E.T. Jorgensen","submitted_at":"2009-03-08T21:06:38Z","abstract_excerpt":"We find conditions under which two measure preserving actions of two groups on the same space have a common fundamental domain. Our results apply to commuting actions with separate fundamental domains, lattices in groups of polynomial growth, and some semidirect products. We prove that two lattices of equal co-volume in a group of polynomial growth, one acting on the left, the other on the right, have a common fundamental domain."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0903.1456","kind":"arxiv","version":3},"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":"2026-05-18T03:33:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"a7Mv6MipgkA3FlMhlDsinWWlUe4on36DnXcQ7gVrOej5mcWaLt0nJec/a9ZAoZ0QN6b/hgxFmxcWxipYfPdEDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T17:41:16.711835Z"},"content_sha256":"6cf51a0ffedd4236a0cc41cc9f4a989cef0e011e6462fa645da4e1dbfe523bb0","schema_version":"1.0","event_id":"sha256:6cf51a0ffedd4236a0cc41cc9f4a989cef0e011e6462fa645da4e1dbfe523bb0"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XNNPBP6KK62KP34WE363OZ4TP6/bundle.json","state_url":"https://pith.science/pith/XNNPBP6KK62KP34WE363OZ4TP6/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XNNPBP6KK62KP34WE363OZ4TP6/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":"2026-06-07T17:41:16Z","links":{"resolver":"https://pith.science/pith/XNNPBP6KK62KP34WE363OZ4TP6","bundle":"https://pith.science/pith/XNNPBP6KK62KP34WE363OZ4TP6/bundle.json","state":"https://pith.science/pith/XNNPBP6KK62KP34WE363OZ4TP6/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XNNPBP6KK62KP34WE363OZ4TP6/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:XNNPBP6KK62KP34WE363OZ4TP6","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":"8464e6ea3bc2954774c6367988c57701cd1adc436e52a6efb4232f02555df877","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2009-03-08T21:06:38Z","title_canon_sha256":"8e5ead1acfcb798356c615091a9ea5060511fcbcad681b3988e0220766938441"},"schema_version":"1.0","source":{"id":"0903.1456","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0903.1456","created_at":"2026-05-18T03:33:27Z"},{"alias_kind":"arxiv_version","alias_value":"0903.1456v3","created_at":"2026-05-18T03:33:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0903.1456","created_at":"2026-05-18T03:33:27Z"},{"alias_kind":"pith_short_12","alias_value":"XNNPBP6KK62K","created_at":"2026-05-18T12:26:02Z"},{"alias_kind":"pith_short_16","alias_value":"XNNPBP6KK62KP34W","created_at":"2026-05-18T12:26:02Z"},{"alias_kind":"pith_short_8","alias_value":"XNNPBP6K","created_at":"2026-05-18T12:26:02Z"}],"graph_snapshots":[{"event_id":"sha256:6cf51a0ffedd4236a0cc41cc9f4a989cef0e011e6462fa645da4e1dbfe523bb0","target":"graph","created_at":"2026-05-18T03:33:27Z","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 find conditions under which two measure preserving actions of two groups on the same space have a common fundamental domain. Our results apply to commuting actions with separate fundamental domains, lattices in groups of polynomial growth, and some semidirect products. We prove that two lattices of equal co-volume in a group of polynomial growth, one acting on the left, the other on the right, have a common fundamental domain.","authors_text":"Deguang Han, Dorin Ervin Dutkay, Gabriel Picioroaga, Palle E.T. Jorgensen","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2009-03-08T21:06:38Z","title":"On common fundamental domains"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0903.1456","kind":"arxiv","version":3},"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:a154b1fb144413666999757dd51c4a008e2edca714aa5407b6cdd71bd67a04d6","target":"record","created_at":"2026-05-18T03:33:27Z","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":"8464e6ea3bc2954774c6367988c57701cd1adc436e52a6efb4232f02555df877","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2009-03-08T21:06:38Z","title_canon_sha256":"8e5ead1acfcb798356c615091a9ea5060511fcbcad681b3988e0220766938441"},"schema_version":"1.0","source":{"id":"0903.1456","kind":"arxiv","version":3}},"canonical_sha256":"bb5af0bfca57b4a7ef9626fdb767937fa020bbbb49990af92994841f6d794188","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bb5af0bfca57b4a7ef9626fdb767937fa020bbbb49990af92994841f6d794188","first_computed_at":"2026-05-18T03:33:27.020183Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:33:27.020183Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"g0EwEy3Edp7cVr5Ic0GQYSu7CFIQs1fE1b1F3tcNAbxgqZ8nGN3aBx1ISm0etW2rMgj2YzvOXZXut0+JGVgLCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:33:27.020968Z","signed_message":"canonical_sha256_bytes"},"source_id":"0903.1456","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a154b1fb144413666999757dd51c4a008e2edca714aa5407b6cdd71bd67a04d6","sha256:6cf51a0ffedd4236a0cc41cc9f4a989cef0e011e6462fa645da4e1dbfe523bb0"],"state_sha256":"5a4d9960b11aa8d29ffea672d397c6c109fbd2d94df950dd6904d6aa9997615a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BpzaeQ74KkXlxls7EMHan/nT7/eqFbnG+wPXuUwUT6Jro4QqUXeCI4+sTKJ6mytdV/7feU40Z5/DctstAtUADA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-07T17:41:16.715799Z","bundle_sha256":"7187262bcfd33f90e4b3c31a4accb151952485d7367dcba0daee87a14a34fe4d"}}