{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:V4VRM6WB3RA2JJEVP7CYWILFIW","short_pith_number":"pith:V4VRM6WB","canonical_record":{"source":{"id":"2402.10869","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DG","submitted_at":"2024-02-16T18:12:05Z","cross_cats_sorted":["math.GR","math.RT"],"title_canon_sha256":"2ffe0cae5f911e921f86c438930218ab9e1e64aa8dea77f6ff2102a1c1789871","abstract_canon_sha256":"8f7246da9eec5d9b1b9b69fe31cbfc3659e832dc40143c3653c53cb76267a79c"},"schema_version":"1.0"},"canonical_sha256":"af2b167ac1dc41a4a4957fc58b216545b95f55da1380faaedf9330146bc1102e","source":{"kind":"arxiv","id":"2402.10869","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2402.10869","created_at":"2026-07-05T07:46:01Z"},{"alias_kind":"arxiv_version","alias_value":"2402.10869v1","created_at":"2026-07-05T07:46:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2402.10869","created_at":"2026-07-05T07:46:01Z"},{"alias_kind":"pith_short_12","alias_value":"V4VRM6WB3RA2","created_at":"2026-07-05T07:46:01Z"},{"alias_kind":"pith_short_16","alias_value":"V4VRM6WB3RA2JJEV","created_at":"2026-07-05T07:46:01Z"},{"alias_kind":"pith_short_8","alias_value":"V4VRM6WB","created_at":"2026-07-05T07:46:01Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:V4VRM6WB3RA2JJEVP7CYWILFIW","target":"record","payload":{"canonical_record":{"source":{"id":"2402.10869","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DG","submitted_at":"2024-02-16T18:12:05Z","cross_cats_sorted":["math.GR","math.RT"],"title_canon_sha256":"2ffe0cae5f911e921f86c438930218ab9e1e64aa8dea77f6ff2102a1c1789871","abstract_canon_sha256":"8f7246da9eec5d9b1b9b69fe31cbfc3659e832dc40143c3653c53cb76267a79c"},"schema_version":"1.0"},"canonical_sha256":"af2b167ac1dc41a4a4957fc58b216545b95f55da1380faaedf9330146bc1102e","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T07:46:01.893030Z","signature_b64":"BoevNvRYWlmQ00w4sHfy0S1a+Twic8fQ7u7QqYr1CdoSdYdSQmuCTRMWSrgxGOGhIUxdIC0hacsl+XUT1C+DAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"af2b167ac1dc41a4a4957fc58b216545b95f55da1380faaedf9330146bc1102e","last_reissued_at":"2026-07-05T07:46:01.892456Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T07:46:01.892456Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2402.10869","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T07:46:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9zeNG+8P4DHGQ1KN2Eim1i5MVXI1khiWY5AUz3fIAMzHOrsOJPGw65BUToWz5x/WvfRxxm8Qoy/Ty+u3zykECg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T17:06:56.380351Z"},"content_sha256":"94975b60cc1331a691fdf2d58cc20d3216a1171e59008fc484f85abfb0bc72fb","schema_version":"1.0","event_id":"sha256:94975b60cc1331a691fdf2d58cc20d3216a1171e59008fc484f85abfb0bc72fb"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:V4VRM6WB3RA2JJEVP7CYWILFIW","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Hyperbolic groups and spherical minimal surfaces","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.GR","math.RT"],"primary_cat":"math.DG","authors_text":"Antoine Song","submitted_at":"2024-02-16T18:12:05Z","abstract_excerpt":"Let $M$ be a closed, oriented, negatively curved, $n$-dimensional manifold with fundamental group $\\Gamma$. Let $S^\\infty$ be the unit sphere in $\\ell^2(\\Gamma)$, on which $\\Gamma$ acts by the regular representation. The spherical volume of $M$ is a topological invariant introduced by Besson-Courtois-Gallot. We show that it is equal to the area of an $n$-dimensional area-minimizing minimal surface inside the ultralimit of $S^\\infty/\\Gamma$, in the sense of Ambrosio-Kirchheim. Our proof combines the theory of metric currents with a study of limits of the regular representation of torsion-free h"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2402.10869","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2402.10869/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T07:46:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KzblClsdQJhve6RqRdiTmAIBl1Vs6QxUav/5cCD+r3jwc9bkohxYa/gLao8EemxTCCqZDNHYWomnNnHyrHBmCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T17:06:56.380737Z"},"content_sha256":"6644eacfa8156d03248ade12f3d4a1833f86c4bd3c5668879c903278220f680d","schema_version":"1.0","event_id":"sha256:6644eacfa8156d03248ade12f3d4a1833f86c4bd3c5668879c903278220f680d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/V4VRM6WB3RA2JJEVP7CYWILFIW/bundle.json","state_url":"https://pith.science/pith/V4VRM6WB3RA2JJEVP7CYWILFIW/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/V4VRM6WB3RA2JJEVP7CYWILFIW/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":"2026-08-13T17:06:56Z","links":{"resolver":"https://pith.science/pith/V4VRM6WB3RA2JJEVP7CYWILFIW","bundle":"https://pith.science/pith/V4VRM6WB3RA2JJEVP7CYWILFIW/bundle.json","state":"https://pith.science/pith/V4VRM6WB3RA2JJEVP7CYWILFIW/state.json","well_known_bundle":"https://pith.science/.well-known/pith/V4VRM6WB3RA2JJEVP7CYWILFIW/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:V4VRM6WB3RA2JJEVP7CYWILFIW","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":"8f7246da9eec5d9b1b9b69fe31cbfc3659e832dc40143c3653c53cb76267a79c","cross_cats_sorted":["math.GR","math.RT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DG","submitted_at":"2024-02-16T18:12:05Z","title_canon_sha256":"2ffe0cae5f911e921f86c438930218ab9e1e64aa8dea77f6ff2102a1c1789871"},"schema_version":"1.0","source":{"id":"2402.10869","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2402.10869","created_at":"2026-07-05T07:46:01Z"},{"alias_kind":"arxiv_version","alias_value":"2402.10869v1","created_at":"2026-07-05T07:46:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2402.10869","created_at":"2026-07-05T07:46:01Z"},{"alias_kind":"pith_short_12","alias_value":"V4VRM6WB3RA2","created_at":"2026-07-05T07:46:01Z"},{"alias_kind":"pith_short_16","alias_value":"V4VRM6WB3RA2JJEV","created_at":"2026-07-05T07:46:01Z"},{"alias_kind":"pith_short_8","alias_value":"V4VRM6WB","created_at":"2026-07-05T07:46:01Z"}],"graph_snapshots":[{"event_id":"sha256:6644eacfa8156d03248ade12f3d4a1833f86c4bd3c5668879c903278220f680d","target":"graph","created_at":"2026-07-05T07:46:01Z","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/2402.10869/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $M$ be a closed, oriented, negatively curved, $n$-dimensional manifold with fundamental group $\\Gamma$. Let $S^\\infty$ be the unit sphere in $\\ell^2(\\Gamma)$, on which $\\Gamma$ acts by the regular representation. The spherical volume of $M$ is a topological invariant introduced by Besson-Courtois-Gallot. We show that it is equal to the area of an $n$-dimensional area-minimizing minimal surface inside the ultralimit of $S^\\infty/\\Gamma$, in the sense of Ambrosio-Kirchheim. Our proof combines the theory of metric currents with a study of limits of the regular representation of torsion-free h","authors_text":"Antoine Song","cross_cats":["math.GR","math.RT"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DG","submitted_at":"2024-02-16T18:12:05Z","title":"Hyperbolic groups and spherical minimal surfaces"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2402.10869","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":"sha256:94975b60cc1331a691fdf2d58cc20d3216a1171e59008fc484f85abfb0bc72fb","target":"record","created_at":"2026-07-05T07:46:01Z","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":"8f7246da9eec5d9b1b9b69fe31cbfc3659e832dc40143c3653c53cb76267a79c","cross_cats_sorted":["math.GR","math.RT"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.DG","submitted_at":"2024-02-16T18:12:05Z","title_canon_sha256":"2ffe0cae5f911e921f86c438930218ab9e1e64aa8dea77f6ff2102a1c1789871"},"schema_version":"1.0","source":{"id":"2402.10869","kind":"arxiv","version":1}},"canonical_sha256":"af2b167ac1dc41a4a4957fc58b216545b95f55da1380faaedf9330146bc1102e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"af2b167ac1dc41a4a4957fc58b216545b95f55da1380faaedf9330146bc1102e","first_computed_at":"2026-07-05T07:46:01.892456Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T07:46:01.892456Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"BoevNvRYWlmQ00w4sHfy0S1a+Twic8fQ7u7QqYr1CdoSdYdSQmuCTRMWSrgxGOGhIUxdIC0hacsl+XUT1C+DAw==","signature_status":"signed_v1","signed_at":"2026-07-05T07:46:01.893030Z","signed_message":"canonical_sha256_bytes"},"source_id":"2402.10869","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:94975b60cc1331a691fdf2d58cc20d3216a1171e59008fc484f85abfb0bc72fb","sha256:6644eacfa8156d03248ade12f3d4a1833f86c4bd3c5668879c903278220f680d"],"state_sha256":"72b16d2fd3b2f6c96b1d2b3c9dca7188ddb4736e9f7186107ff5dc2f439d13c7"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"sNGAp/13Uyy9Xk5bhTCOhMl3bEEFLEYilnS/+XDc9LRC6ASHGBpKaShdEJnxBVLKWgnsXX04ZMAZTELVOwbqBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-13T17:06:56.385099Z","bundle_sha256":"53b2714da4e23f84ac5f879000e563f2ab404a3f795fd536dc8aa7df3fb40364"}}