{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:DB53VZOLXHSPVY2EMR2FZZZVYT","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":"98bc51be7156ec9fed24d2fb2f0ddf977ef507c8fced7f6f161496b5b80cb15a","cross_cats_sorted":["math.GR","math.MG"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DG","submitted_at":"2024-10-09T15:00:02Z","title_canon_sha256":"a5d30b690fc917b6d885423ddf4455b36f9f3e6074a582eb861117a4f9c6342b"},"schema_version":"1.0","source":{"id":"2410.07291","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2410.07291","created_at":"2026-07-05T09:18:34Z"},{"alias_kind":"arxiv_version","alias_value":"2410.07291v1","created_at":"2026-07-05T09:18:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2410.07291","created_at":"2026-07-05T09:18:34Z"},{"alias_kind":"pith_short_12","alias_value":"DB53VZOLXHSP","created_at":"2026-07-05T09:18:34Z"},{"alias_kind":"pith_short_16","alias_value":"DB53VZOLXHSPVY2E","created_at":"2026-07-05T09:18:34Z"},{"alias_kind":"pith_short_8","alias_value":"DB53VZOL","created_at":"2026-07-05T09:18:34Z"}],"graph_snapshots":[{"event_id":"sha256:8ff675998ea167e33682c4fe4ec107f24965a8e6a45a55334a80ffc8e27fcb14","target":"graph","created_at":"2026-07-05T09:18:34Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2410.07291/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"This book explores geometries defined by left-invariant distance functions on Lie groups, with a particular focus on nilpotent groups and Carnot groups equipped with geodesic distances. Geodesic left-invariant metrics are either sub-Riemannian or their generalizations, known as sub-Finsler geometries or Carnot-Carath\\'eodory metrics. The primary objective is to illustrate how these non-smooth geometries, together with a Lie group structure, manifest in various mathematical fields, including metric geometry and geometric group theory.\n  Additionally, the book demonstrates the role of metric Lie","authors_text":"Enrico Le Donne","cross_cats":["math.GR","math.MG"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DG","submitted_at":"2024-10-09T15:00:02Z","title":"Metric Lie Groups. Carnot-Carath\\'eodory spaces from the homogeneous viewpoint"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2410.07291","kind":"arxiv","version":1},"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":"sha256:6aaa08f6642878d5d4b87ef7978bb08a1a951c75e06d3b0abb2a4fd40d47195c","target":"record","created_at":"2026-07-05T09:18:34Z","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":"98bc51be7156ec9fed24d2fb2f0ddf977ef507c8fced7f6f161496b5b80cb15a","cross_cats_sorted":["math.GR","math.MG"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DG","submitted_at":"2024-10-09T15:00:02Z","title_canon_sha256":"a5d30b690fc917b6d885423ddf4455b36f9f3e6074a582eb861117a4f9c6342b"},"schema_version":"1.0","source":{"id":"2410.07291","kind":"arxiv","version":1}},"canonical_sha256":"187bbae5cbb9e4fae34464745ce735c4fc0437ba7cdf9fd058398e1bc7165862","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"187bbae5cbb9e4fae34464745ce735c4fc0437ba7cdf9fd058398e1bc7165862","first_computed_at":"2026-07-05T09:18:34.062948Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T09:18:34.062948Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"rd76DB/Ztk4c88z1nyYFPY+z9tCfj3cZqGvlHl7c6HBzhdIQX1KTlVRzayN3N/+P6KQAILqhmW3TGrbqgAtGCw==","signature_status":"signed_v1","signed_at":"2026-07-05T09:18:34.063409Z","signed_message":"canonical_sha256_bytes"},"source_id":"2410.07291","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6aaa08f6642878d5d4b87ef7978bb08a1a951c75e06d3b0abb2a4fd40d47195c","sha256:8ff675998ea167e33682c4fe4ec107f24965a8e6a45a55334a80ffc8e27fcb14"],"state_sha256":"6137223a921fbed778acbf94da07ce34ff9f4f57b939f655e96aa5f5b62f5932"}