{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:TJPI2PHX7CSHFIFZB7MT4HXN3V","short_pith_number":"pith:TJPI2PHX","canonical_record":{"source":{"id":"1401.3611","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2014-01-15T14:43:21Z","cross_cats_sorted":["math.FA","math.MG"],"title_canon_sha256":"1a9f9795719d726878ec0f66616ca6cac3608e4ef10e6932d2768a7c3cc1f249","abstract_canon_sha256":"b0f27f0f82bc30bc7a01af7ff1efb8a4e0b69a60abe3c1d517c41e51bce79f83"},"schema_version":"1.0"},"canonical_sha256":"9a5e8d3cf7f8a472a0b90fd93e1eeddd5c36630508c2295c1212b50857d81070","source":{"kind":"arxiv","id":"1401.3611","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1401.3611","created_at":"2026-05-18T01:25:39Z"},{"alias_kind":"arxiv_version","alias_value":"1401.3611v2","created_at":"2026-05-18T01:25:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1401.3611","created_at":"2026-05-18T01:25:39Z"},{"alias_kind":"pith_short_12","alias_value":"TJPI2PHX7CSH","created_at":"2026-05-18T12:28:49Z"},{"alias_kind":"pith_short_16","alias_value":"TJPI2PHX7CSHFIFZ","created_at":"2026-05-18T12:28:49Z"},{"alias_kind":"pith_short_8","alias_value":"TJPI2PHX","created_at":"2026-05-18T12:28:49Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:TJPI2PHX7CSHFIFZB7MT4HXN3V","target":"record","payload":{"canonical_record":{"source":{"id":"1401.3611","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2014-01-15T14:43:21Z","cross_cats_sorted":["math.FA","math.MG"],"title_canon_sha256":"1a9f9795719d726878ec0f66616ca6cac3608e4ef10e6932d2768a7c3cc1f249","abstract_canon_sha256":"b0f27f0f82bc30bc7a01af7ff1efb8a4e0b69a60abe3c1d517c41e51bce79f83"},"schema_version":"1.0"},"canonical_sha256":"9a5e8d3cf7f8a472a0b90fd93e1eeddd5c36630508c2295c1212b50857d81070","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:25:39.904508Z","signature_b64":"1kaEjjSwBTj6YbaBa2L3efl5MsvvYcXO2Kc2yYCNTroakQAnYHp2CRTQue4fuCR4t0sKTbwf/zpU/wycLG+xCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9a5e8d3cf7f8a472a0b90fd93e1eeddd5c36630508c2295c1212b50857d81070","last_reissued_at":"2026-05-18T01:25:39.904002Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:25:39.904002Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1401.3611","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":"2026-05-18T01:25:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VinQG8h/8hKKLji/efW74WXJ8ZaOTBCZu6TM7UnGLYI+CENIoVxKDvKcurd/0JNNiYyInslfw7CkEw6lKSEBAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T21:34:13.735115Z"},"content_sha256":"62cad25a0b9ed94b64d163215e73bc09ae9d6480c9ad9ae5d2ed003c2040ce21","schema_version":"1.0","event_id":"sha256:62cad25a0b9ed94b64d163215e73bc09ae9d6480c9ad9ae5d2ed003c2040ce21"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:TJPI2PHX7CSHFIFZB7MT4HXN3V","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Strong property (T) for higher rank simple Lie groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA","math.MG"],"primary_cat":"math.GR","authors_text":"Mikael de la Salle, Tim de Laat","submitted_at":"2014-01-15T14:43:21Z","abstract_excerpt":"We prove that connected higher rank simple Lie groups have Lafforgue's strong property (T) with respect to a certain class of Banach spaces $\\mathcal{E}_{10}$ containing many classical superreflexive spaces and some non-reflexive spaces as well. This generalizes the result of Lafforgue asserting that $\\mathrm{SL}(3,\\mathbb{R})$ has strong property (T) with respect to Hilbert spaces and the more recent result of the second named author asserting that $\\mathrm{SL}(3,\\mathbb{R})$ has strong property (T) with respect to a certain larger class of Banach spaces. For the generalization to higher rank"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.3611","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":"2026-05-18T01:25:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"no8cPjUNRVXWZ3vTmkUB1w++7UGZOfJKJmQg6fPTYwZd4OF7RBy+g2tM673kYaEmbCcm0jD8J/8jEvPpGzDZCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T21:34:13.735802Z"},"content_sha256":"754b543efbfd675e8e0a6d29fd464d7722b81bf95ccc6a105fbe5b04fc07076e","schema_version":"1.0","event_id":"sha256:754b543efbfd675e8e0a6d29fd464d7722b81bf95ccc6a105fbe5b04fc07076e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/TJPI2PHX7CSHFIFZB7MT4HXN3V/bundle.json","state_url":"https://pith.science/pith/TJPI2PHX7CSHFIFZB7MT4HXN3V/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/TJPI2PHX7CSHFIFZB7MT4HXN3V/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":"2026-05-30T21:34:13Z","links":{"resolver":"https://pith.science/pith/TJPI2PHX7CSHFIFZB7MT4HXN3V","bundle":"https://pith.science/pith/TJPI2PHX7CSHFIFZB7MT4HXN3V/bundle.json","state":"https://pith.science/pith/TJPI2PHX7CSHFIFZB7MT4HXN3V/state.json","well_known_bundle":"https://pith.science/.well-known/pith/TJPI2PHX7CSHFIFZB7MT4HXN3V/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:TJPI2PHX7CSHFIFZB7MT4HXN3V","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":"b0f27f0f82bc30bc7a01af7ff1efb8a4e0b69a60abe3c1d517c41e51bce79f83","cross_cats_sorted":["math.FA","math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2014-01-15T14:43:21Z","title_canon_sha256":"1a9f9795719d726878ec0f66616ca6cac3608e4ef10e6932d2768a7c3cc1f249"},"schema_version":"1.0","source":{"id":"1401.3611","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1401.3611","created_at":"2026-05-18T01:25:39Z"},{"alias_kind":"arxiv_version","alias_value":"1401.3611v2","created_at":"2026-05-18T01:25:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1401.3611","created_at":"2026-05-18T01:25:39Z"},{"alias_kind":"pith_short_12","alias_value":"TJPI2PHX7CSH","created_at":"2026-05-18T12:28:49Z"},{"alias_kind":"pith_short_16","alias_value":"TJPI2PHX7CSHFIFZ","created_at":"2026-05-18T12:28:49Z"},{"alias_kind":"pith_short_8","alias_value":"TJPI2PHX","created_at":"2026-05-18T12:28:49Z"}],"graph_snapshots":[{"event_id":"sha256:754b543efbfd675e8e0a6d29fd464d7722b81bf95ccc6a105fbe5b04fc07076e","target":"graph","created_at":"2026-05-18T01:25:39Z","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 prove that connected higher rank simple Lie groups have Lafforgue's strong property (T) with respect to a certain class of Banach spaces $\\mathcal{E}_{10}$ containing many classical superreflexive spaces and some non-reflexive spaces as well. This generalizes the result of Lafforgue asserting that $\\mathrm{SL}(3,\\mathbb{R})$ has strong property (T) with respect to Hilbert spaces and the more recent result of the second named author asserting that $\\mathrm{SL}(3,\\mathbb{R})$ has strong property (T) with respect to a certain larger class of Banach spaces. For the generalization to higher rank","authors_text":"Mikael de la Salle, Tim de Laat","cross_cats":["math.FA","math.MG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2014-01-15T14:43:21Z","title":"Strong property (T) for higher rank simple Lie groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.3611","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":"sha256:62cad25a0b9ed94b64d163215e73bc09ae9d6480c9ad9ae5d2ed003c2040ce21","target":"record","created_at":"2026-05-18T01:25:39Z","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":"b0f27f0f82bc30bc7a01af7ff1efb8a4e0b69a60abe3c1d517c41e51bce79f83","cross_cats_sorted":["math.FA","math.MG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2014-01-15T14:43:21Z","title_canon_sha256":"1a9f9795719d726878ec0f66616ca6cac3608e4ef10e6932d2768a7c3cc1f249"},"schema_version":"1.0","source":{"id":"1401.3611","kind":"arxiv","version":2}},"canonical_sha256":"9a5e8d3cf7f8a472a0b90fd93e1eeddd5c36630508c2295c1212b50857d81070","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9a5e8d3cf7f8a472a0b90fd93e1eeddd5c36630508c2295c1212b50857d81070","first_computed_at":"2026-05-18T01:25:39.904002Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:25:39.904002Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"1kaEjjSwBTj6YbaBa2L3efl5MsvvYcXO2Kc2yYCNTroakQAnYHp2CRTQue4fuCR4t0sKTbwf/zpU/wycLG+xCg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:25:39.904508Z","signed_message":"canonical_sha256_bytes"},"source_id":"1401.3611","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:62cad25a0b9ed94b64d163215e73bc09ae9d6480c9ad9ae5d2ed003c2040ce21","sha256:754b543efbfd675e8e0a6d29fd464d7722b81bf95ccc6a105fbe5b04fc07076e"],"state_sha256":"0dcbfb37d1b3020f97238c85ca279c7d5fa8c19a018aa65da94749038731897f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"v2X/1YWjqlmhjX6vYK4pEHR8T0K6k2w64JdtI/wIt8MUhTjLX2NoPgsx78GHJwXUVkoYbNuNdOP9TsK1b91SDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T21:34:13.739266Z","bundle_sha256":"6ba0de57c2633546ae48072a3b15207c8b93c45b23b1c8831f13727a152bf980"}}