{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:E6EDVY6KR3EU6IDCPWIJVOKFUQ","short_pith_number":"pith:E6EDVY6K","canonical_record":{"source":{"id":"1612.06220","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-12-19T15:14:18Z","cross_cats_sorted":[],"title_canon_sha256":"e6ed09c767661b45f079f6a72f1a526b6fc1ea35b353c8d8362cf973d6389b6c","abstract_canon_sha256":"3a6fd03596267c8f48fa43cff4667ec72ba8b9295732959cc183323b65c69040"},"schema_version":"1.0"},"canonical_sha256":"27883ae3ca8ec94f20627d909ab945a412b907eac257665623a758fd487bfea0","source":{"kind":"arxiv","id":"1612.06220","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1612.06220","created_at":"2026-05-18T00:03:57Z"},{"alias_kind":"arxiv_version","alias_value":"1612.06220v3","created_at":"2026-05-18T00:03:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.06220","created_at":"2026-05-18T00:03:57Z"},{"alias_kind":"pith_short_12","alias_value":"E6EDVY6KR3EU","created_at":"2026-05-18T12:30:12Z"},{"alias_kind":"pith_short_16","alias_value":"E6EDVY6KR3EU6IDC","created_at":"2026-05-18T12:30:12Z"},{"alias_kind":"pith_short_8","alias_value":"E6EDVY6K","created_at":"2026-05-18T12:30:12Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:E6EDVY6KR3EU6IDCPWIJVOKFUQ","target":"record","payload":{"canonical_record":{"source":{"id":"1612.06220","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-12-19T15:14:18Z","cross_cats_sorted":[],"title_canon_sha256":"e6ed09c767661b45f079f6a72f1a526b6fc1ea35b353c8d8362cf973d6389b6c","abstract_canon_sha256":"3a6fd03596267c8f48fa43cff4667ec72ba8b9295732959cc183323b65c69040"},"schema_version":"1.0"},"canonical_sha256":"27883ae3ca8ec94f20627d909ab945a412b907eac257665623a758fd487bfea0","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:03:57.188078Z","signature_b64":"OCxRAji+Mx1vmWadcojU/TWZ54q66EQcA6hshu+xhSPqQN7ZMBej7bbsH18ZQROkFBkCOCFDZqYOo8K/zLfVDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"27883ae3ca8ec94f20627d909ab945a412b907eac257665623a758fd487bfea0","last_reissued_at":"2026-05-18T00:03:57.187512Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:03:57.187512Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1612.06220","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":"2026-05-18T00:03:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VlzORjDkQT9AdtJ1zT3okIwtdZtl08TDOUZqFGO8ynbNhSfjV+a/Bvk9Wn/e4FdUregHh1p5bZKyhWT3APNjCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T18:30:28.052796Z"},"content_sha256":"b5de25e54d836c6b1c093e636066daf69fb16ce555b8b1df0cc2a6c93479e131","schema_version":"1.0","event_id":"sha256:b5de25e54d836c6b1c093e636066daf69fb16ce555b8b1df0cc2a6c93479e131"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:E6EDVY6KR3EU6IDCPWIJVOKFUQ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Lattices in amenable groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"P-E. Caprace, Sh. Mozes, T. Gelander, U. Bader","submitted_at":"2016-12-19T15:14:18Z","abstract_excerpt":"Let $G$ be a locally compact amenable group. We say that G has property (M) if every closed subgroup of finite covolume in G is cocompact. A classical theorem of Mostow ensures that connected solvable Lie groups have property (M). We prove a non-Archimedean extension of Mostow's theorem by showing the amenable linear locally compact groups have property (M). However property (M) does not hold for all solvable locally compact groups: indeed, we exhibit an example of a metabelian locally compact group with a non-uniform lattice. We show that compactly generated metabelian groups, and more genera"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.06220","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":"2026-05-18T00:03:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uOjeuis+GYTz9xcFJ+u4OLmNojaNp6BZnGOsezILwx23bkpx2rIC3e4fXTAUr60dfqvHeWn0Ce2VLRfNrzWUDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T18:30:28.053529Z"},"content_sha256":"44d2a9c5c27d4694310e94d5316197992d8ab2ee74672833ad46e2c8450ad6d3","schema_version":"1.0","event_id":"sha256:44d2a9c5c27d4694310e94d5316197992d8ab2ee74672833ad46e2c8450ad6d3"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/E6EDVY6KR3EU6IDCPWIJVOKFUQ/bundle.json","state_url":"https://pith.science/pith/E6EDVY6KR3EU6IDCPWIJVOKFUQ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/E6EDVY6KR3EU6IDCPWIJVOKFUQ/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":"2026-06-08T18:30:28Z","links":{"resolver":"https://pith.science/pith/E6EDVY6KR3EU6IDCPWIJVOKFUQ","bundle":"https://pith.science/pith/E6EDVY6KR3EU6IDCPWIJVOKFUQ/bundle.json","state":"https://pith.science/pith/E6EDVY6KR3EU6IDCPWIJVOKFUQ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/E6EDVY6KR3EU6IDCPWIJVOKFUQ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:E6EDVY6KR3EU6IDCPWIJVOKFUQ","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":"3a6fd03596267c8f48fa43cff4667ec72ba8b9295732959cc183323b65c69040","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-12-19T15:14:18Z","title_canon_sha256":"e6ed09c767661b45f079f6a72f1a526b6fc1ea35b353c8d8362cf973d6389b6c"},"schema_version":"1.0","source":{"id":"1612.06220","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1612.06220","created_at":"2026-05-18T00:03:57Z"},{"alias_kind":"arxiv_version","alias_value":"1612.06220v3","created_at":"2026-05-18T00:03:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.06220","created_at":"2026-05-18T00:03:57Z"},{"alias_kind":"pith_short_12","alias_value":"E6EDVY6KR3EU","created_at":"2026-05-18T12:30:12Z"},{"alias_kind":"pith_short_16","alias_value":"E6EDVY6KR3EU6IDC","created_at":"2026-05-18T12:30:12Z"},{"alias_kind":"pith_short_8","alias_value":"E6EDVY6K","created_at":"2026-05-18T12:30:12Z"}],"graph_snapshots":[{"event_id":"sha256:44d2a9c5c27d4694310e94d5316197992d8ab2ee74672833ad46e2c8450ad6d3","target":"graph","created_at":"2026-05-18T00:03:57Z","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":"Let $G$ be a locally compact amenable group. We say that G has property (M) if every closed subgroup of finite covolume in G is cocompact. A classical theorem of Mostow ensures that connected solvable Lie groups have property (M). We prove a non-Archimedean extension of Mostow's theorem by showing the amenable linear locally compact groups have property (M). However property (M) does not hold for all solvable locally compact groups: indeed, we exhibit an example of a metabelian locally compact group with a non-uniform lattice. We show that compactly generated metabelian groups, and more genera","authors_text":"P-E. Caprace, Sh. Mozes, T. Gelander, U. Bader","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-12-19T15:14:18Z","title":"Lattices in amenable groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.06220","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:b5de25e54d836c6b1c093e636066daf69fb16ce555b8b1df0cc2a6c93479e131","target":"record","created_at":"2026-05-18T00:03:57Z","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":"3a6fd03596267c8f48fa43cff4667ec72ba8b9295732959cc183323b65c69040","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-12-19T15:14:18Z","title_canon_sha256":"e6ed09c767661b45f079f6a72f1a526b6fc1ea35b353c8d8362cf973d6389b6c"},"schema_version":"1.0","source":{"id":"1612.06220","kind":"arxiv","version":3}},"canonical_sha256":"27883ae3ca8ec94f20627d909ab945a412b907eac257665623a758fd487bfea0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"27883ae3ca8ec94f20627d909ab945a412b907eac257665623a758fd487bfea0","first_computed_at":"2026-05-18T00:03:57.187512Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:03:57.187512Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"OCxRAji+Mx1vmWadcojU/TWZ54q66EQcA6hshu+xhSPqQN7ZMBej7bbsH18ZQROkFBkCOCFDZqYOo8K/zLfVDA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:03:57.188078Z","signed_message":"canonical_sha256_bytes"},"source_id":"1612.06220","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b5de25e54d836c6b1c093e636066daf69fb16ce555b8b1df0cc2a6c93479e131","sha256:44d2a9c5c27d4694310e94d5316197992d8ab2ee74672833ad46e2c8450ad6d3"],"state_sha256":"32cf1b10c67f8ec372dd078c12f72e7d2f42d3cd73287292862f16fd59801444"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JhG9j8UX8SnQt6jtyAZ0RHu+40NUaRYH+07EQu0tQ+14UkZq/l6WD05Tp3416vAZdvQPNEF3xpp07eUF8fY9BQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T18:30:28.057407Z","bundle_sha256":"f49ae1ed8ccacc35bb4064242473cc306fd3c6820ae16c75037404b50e1173d4"}}