{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:7RFB3A6FPPLC2UMUZNFDGJWZYH","short_pith_number":"pith:7RFB3A6F","canonical_record":{"source":{"id":"1409.8041","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2014-09-29T09:28:41Z","cross_cats_sorted":[],"title_canon_sha256":"cfd66894f8b13a7b602001e3a976696617ee1a555428fc06bf7ab611a615c06c","abstract_canon_sha256":"0b8b4ca1662c5a66084d58c895761af9d9e6850c3d16e14264dad2581d263917"},"schema_version":"1.0"},"canonical_sha256":"fc4a1d83c57bd62d5194cb4a3326d9c1d52b1794a6d796d879b2c7b9156bc49b","source":{"kind":"arxiv","id":"1409.8041","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1409.8041","created_at":"2026-05-18T01:20:29Z"},{"alias_kind":"arxiv_version","alias_value":"1409.8041v2","created_at":"2026-05-18T01:20:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1409.8041","created_at":"2026-05-18T01:20:29Z"},{"alias_kind":"pith_short_12","alias_value":"7RFB3A6FPPLC","created_at":"2026-05-18T12:28:19Z"},{"alias_kind":"pith_short_16","alias_value":"7RFB3A6FPPLC2UMU","created_at":"2026-05-18T12:28:19Z"},{"alias_kind":"pith_short_8","alias_value":"7RFB3A6F","created_at":"2026-05-18T12:28:19Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:7RFB3A6FPPLC2UMUZNFDGJWZYH","target":"record","payload":{"canonical_record":{"source":{"id":"1409.8041","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2014-09-29T09:28:41Z","cross_cats_sorted":[],"title_canon_sha256":"cfd66894f8b13a7b602001e3a976696617ee1a555428fc06bf7ab611a615c06c","abstract_canon_sha256":"0b8b4ca1662c5a66084d58c895761af9d9e6850c3d16e14264dad2581d263917"},"schema_version":"1.0"},"canonical_sha256":"fc4a1d83c57bd62d5194cb4a3326d9c1d52b1794a6d796d879b2c7b9156bc49b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:20:29.878478Z","signature_b64":"s7Ig7CTUWxWHY47dPgEue9AUjryp6lK0EMKjB7gnv6TmdaBE6Xm9d+jGBAzQHDR5isg16cRJZKUvtqBxNkWvDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fc4a1d83c57bd62d5194cb4a3326d9c1d52b1794a6d796d879b2c7b9156bc49b","last_reissued_at":"2026-05-18T01:20:29.877857Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:20:29.877857Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1409.8041","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":"2026-05-18T01:20:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8kF2hDTtzTDLIP+oqzmwPHIUIPjRZtABVXhQnzEdp0sSNqKwBjcHGvXuB6XJgnE2MF4iMm1+KW/hd6qEYwwSDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T03:37:44.701720Z"},"content_sha256":"53c8267ed949ca7b2e0558197d64f70a9fc2c6d822f023dffc9285de877cb3fe","schema_version":"1.0","event_id":"sha256:53c8267ed949ca7b2e0558197d64f70a9fc2c6d822f023dffc9285de877cb3fe"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:7RFB3A6FPPLC2UMUZNFDGJWZYH","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Locally compact lacunary hyperbolic groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Adrien Le Boudec","submitted_at":"2014-09-29T09:28:41Z","abstract_excerpt":"We investigate the class of locally compact lacunary hyperbolic groups. We prove that if a locally compact compactly generated group G admits one asymptotic cone that is a real tree and whose natural transitive isometric action is focal, then G must be a focal hyperbolic group. As an application, we characterize connected Lie groups and linear algebraic groups over an ultrametric local field of characteristic zero having cut-points in one asymptotic cone.\n  We prove several results for locally compact lacunary hyperbolic groups, and extend the characterization of finitely generated lacunary hy"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.8041","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":"2026-05-18T01:20:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"EDATi5ucPbWYhFbXR5IZWkNjB94ORADVXASaHiWKc8iGe/0zHT8GkM3UEEFJOJGQ/kVJYTKMkliBVYQYCKoOAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T03:37:44.702072Z"},"content_sha256":"325a551a77a6eb19f9496e840b5cc77d85d5be224e360696267dde54d095bc23","schema_version":"1.0","event_id":"sha256:325a551a77a6eb19f9496e840b5cc77d85d5be224e360696267dde54d095bc23"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7RFB3A6FPPLC2UMUZNFDGJWZYH/bundle.json","state_url":"https://pith.science/pith/7RFB3A6FPPLC2UMUZNFDGJWZYH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7RFB3A6FPPLC2UMUZNFDGJWZYH/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":"2026-06-03T03:37:44Z","links":{"resolver":"https://pith.science/pith/7RFB3A6FPPLC2UMUZNFDGJWZYH","bundle":"https://pith.science/pith/7RFB3A6FPPLC2UMUZNFDGJWZYH/bundle.json","state":"https://pith.science/pith/7RFB3A6FPPLC2UMUZNFDGJWZYH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7RFB3A6FPPLC2UMUZNFDGJWZYH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:7RFB3A6FPPLC2UMUZNFDGJWZYH","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":"0b8b4ca1662c5a66084d58c895761af9d9e6850c3d16e14264dad2581d263917","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2014-09-29T09:28:41Z","title_canon_sha256":"cfd66894f8b13a7b602001e3a976696617ee1a555428fc06bf7ab611a615c06c"},"schema_version":"1.0","source":{"id":"1409.8041","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1409.8041","created_at":"2026-05-18T01:20:29Z"},{"alias_kind":"arxiv_version","alias_value":"1409.8041v2","created_at":"2026-05-18T01:20:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1409.8041","created_at":"2026-05-18T01:20:29Z"},{"alias_kind":"pith_short_12","alias_value":"7RFB3A6FPPLC","created_at":"2026-05-18T12:28:19Z"},{"alias_kind":"pith_short_16","alias_value":"7RFB3A6FPPLC2UMU","created_at":"2026-05-18T12:28:19Z"},{"alias_kind":"pith_short_8","alias_value":"7RFB3A6F","created_at":"2026-05-18T12:28:19Z"}],"graph_snapshots":[{"event_id":"sha256:325a551a77a6eb19f9496e840b5cc77d85d5be224e360696267dde54d095bc23","target":"graph","created_at":"2026-05-18T01:20:29Z","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 investigate the class of locally compact lacunary hyperbolic groups. We prove that if a locally compact compactly generated group G admits one asymptotic cone that is a real tree and whose natural transitive isometric action is focal, then G must be a focal hyperbolic group. As an application, we characterize connected Lie groups and linear algebraic groups over an ultrametric local field of characteristic zero having cut-points in one asymptotic cone.\n  We prove several results for locally compact lacunary hyperbolic groups, and extend the characterization of finitely generated lacunary hy","authors_text":"Adrien Le Boudec","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2014-09-29T09:28:41Z","title":"Locally compact lacunary hyperbolic groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.8041","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":"sha256:53c8267ed949ca7b2e0558197d64f70a9fc2c6d822f023dffc9285de877cb3fe","target":"record","created_at":"2026-05-18T01:20:29Z","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":"0b8b4ca1662c5a66084d58c895761af9d9e6850c3d16e14264dad2581d263917","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2014-09-29T09:28:41Z","title_canon_sha256":"cfd66894f8b13a7b602001e3a976696617ee1a555428fc06bf7ab611a615c06c"},"schema_version":"1.0","source":{"id":"1409.8041","kind":"arxiv","version":2}},"canonical_sha256":"fc4a1d83c57bd62d5194cb4a3326d9c1d52b1794a6d796d879b2c7b9156bc49b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fc4a1d83c57bd62d5194cb4a3326d9c1d52b1794a6d796d879b2c7b9156bc49b","first_computed_at":"2026-05-18T01:20:29.877857Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:20:29.877857Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"s7Ig7CTUWxWHY47dPgEue9AUjryp6lK0EMKjB7gnv6TmdaBE6Xm9d+jGBAzQHDR5isg16cRJZKUvtqBxNkWvDg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:20:29.878478Z","signed_message":"canonical_sha256_bytes"},"source_id":"1409.8041","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:53c8267ed949ca7b2e0558197d64f70a9fc2c6d822f023dffc9285de877cb3fe","sha256:325a551a77a6eb19f9496e840b5cc77d85d5be224e360696267dde54d095bc23"],"state_sha256":"14d3d7131a0262e089d1bc2aa3778f7f2d75784c9876a9760a21ce9fdc46a7ef"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"pe/iv+SHVn05yVSE/jiwlJ5tWMLT1Kg6/Vs+d+M/QThgnsckPmg588sBrIjGi2TyCTpxYwm6ZLcMhJABXPuGAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T03:37:44.704369Z","bundle_sha256":"16eeba7fba5b48be9d5ddb2211b5b0916833dccec233129b338052dd283d03ce"}}