{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:G25R7SQYEJ5Y7WKHXQ2YGYY4CU","short_pith_number":"pith:G25R7SQY","canonical_record":{"source":{"id":"1210.2961","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2012-10-10T15:59:24Z","cross_cats_sorted":["math.DG","math.GR","math.KT"],"title_canon_sha256":"92b79eb3aa09a053a207304dce6950ca0551fa11859c08a7abbaf736eaa88ce1","abstract_canon_sha256":"9b74a552db6cb2dbe13b4f1733523f6be53afded5e2f0e2d22a999313796b9e4"},"schema_version":"1.0"},"canonical_sha256":"36bb1fca18227b8fd947bc3583631c151946428be1df0ad35d7068e83bfd9c76","source":{"kind":"arxiv","id":"1210.2961","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1210.2961","created_at":"2026-05-18T00:53:40Z"},{"alias_kind":"arxiv_version","alias_value":"1210.2961v4","created_at":"2026-05-18T00:53:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.2961","created_at":"2026-05-18T00:53:40Z"},{"alias_kind":"pith_short_12","alias_value":"G25R7SQYEJ5Y","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_16","alias_value":"G25R7SQYEJ5Y7WKH","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_8","alias_value":"G25R7SQY","created_at":"2026-05-18T12:27:06Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:G25R7SQYEJ5Y7WKHXQ2YGYY4CU","target":"record","payload":{"canonical_record":{"source":{"id":"1210.2961","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2012-10-10T15:59:24Z","cross_cats_sorted":["math.DG","math.GR","math.KT"],"title_canon_sha256":"92b79eb3aa09a053a207304dce6950ca0551fa11859c08a7abbaf736eaa88ce1","abstract_canon_sha256":"9b74a552db6cb2dbe13b4f1733523f6be53afded5e2f0e2d22a999313796b9e4"},"schema_version":"1.0"},"canonical_sha256":"36bb1fca18227b8fd947bc3583631c151946428be1df0ad35d7068e83bfd9c76","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:53:40.987029Z","signature_b64":"RcUAoUjvQGbZZKYmRWT1Xy0Np3ZVBB5uBRlQ166G9zwHYaOxmCFPEASPR7VSfpstriwfKUaDl7WubH9nbWHPDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"36bb1fca18227b8fd947bc3583631c151946428be1df0ad35d7068e83bfd9c76","last_reissued_at":"2026-05-18T00:53:40.986475Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:53:40.986475Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1210.2961","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:53:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"y2Nxzou9VALXJ5VTQ240CDwTXqlX3vQFGrZmpKMj7MsG/nWOI8oRcoD4fHRxwXXboPYBJWFXcqIEo2btP+6gDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T20:37:33.409622Z"},"content_sha256":"944a29b6b3a12354bae682a89839f5225d6bb5942892eeca9defaf812ca0637d","schema_version":"1.0","event_id":"sha256:944a29b6b3a12354bae682a89839f5225d6bb5942892eeca9defaf812ca0637d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:G25R7SQYEJ5Y7WKHXQ2YGYY4CU","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the growth of $L^2$-invariants for sequences of lattices in Lie groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG","math.GR","math.KT"],"primary_cat":"math.RT","authors_text":"Ian Biringer, Iddo Samet, Jean Raimbault, Miklos Abert, Nicolas Bergeron, Nikolay Nikolov, Tsachik Gelander","submitted_at":"2012-10-10T15:59:24Z","abstract_excerpt":"We study the asymptotic behaviour of Betti numbers, twisted torsion and other spectral invariants of sequences of locally symmetric spaces. Our main results are uniform versions of the DeGeorge--Wallach Theorem, of a theorem of Delorme and various other limit multiplicity theorems.\n  A basic idea is to adapt the notion of Benjamini--Schramm convergence (BS-convergence), originally introduced for sequences of finite graphs of bounded degree, to sequences of Riemannian manifolds, and analyze the possible limits. We show that BS-convergence of locally symmetric spaces implies convergence, in an a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.2961","kind":"arxiv","version":4},"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":"2026-05-18T00:53:40Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6nQaTY8xHeoN/DNdSkx1//9p9c2q2kO+5uMHrCR/FkZ8NQ39PbLPZ2mpxaOKg9qlr9dYU0YzVGQMjfpnb++/AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T20:37:33.409964Z"},"content_sha256":"b6e8f5b18b70276993fc224a5d578cbe45e78e7191539c5dd594156e4aab53aa","schema_version":"1.0","event_id":"sha256:b6e8f5b18b70276993fc224a5d578cbe45e78e7191539c5dd594156e4aab53aa"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/G25R7SQYEJ5Y7WKHXQ2YGYY4CU/bundle.json","state_url":"https://pith.science/pith/G25R7SQYEJ5Y7WKHXQ2YGYY4CU/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/G25R7SQYEJ5Y7WKHXQ2YGYY4CU/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":"2026-06-01T20:37:33Z","links":{"resolver":"https://pith.science/pith/G25R7SQYEJ5Y7WKHXQ2YGYY4CU","bundle":"https://pith.science/pith/G25R7SQYEJ5Y7WKHXQ2YGYY4CU/bundle.json","state":"https://pith.science/pith/G25R7SQYEJ5Y7WKHXQ2YGYY4CU/state.json","well_known_bundle":"https://pith.science/.well-known/pith/G25R7SQYEJ5Y7WKHXQ2YGYY4CU/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:G25R7SQYEJ5Y7WKHXQ2YGYY4CU","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":"9b74a552db6cb2dbe13b4f1733523f6be53afded5e2f0e2d22a999313796b9e4","cross_cats_sorted":["math.DG","math.GR","math.KT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2012-10-10T15:59:24Z","title_canon_sha256":"92b79eb3aa09a053a207304dce6950ca0551fa11859c08a7abbaf736eaa88ce1"},"schema_version":"1.0","source":{"id":"1210.2961","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1210.2961","created_at":"2026-05-18T00:53:40Z"},{"alias_kind":"arxiv_version","alias_value":"1210.2961v4","created_at":"2026-05-18T00:53:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1210.2961","created_at":"2026-05-18T00:53:40Z"},{"alias_kind":"pith_short_12","alias_value":"G25R7SQYEJ5Y","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_16","alias_value":"G25R7SQYEJ5Y7WKH","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_8","alias_value":"G25R7SQY","created_at":"2026-05-18T12:27:06Z"}],"graph_snapshots":[{"event_id":"sha256:b6e8f5b18b70276993fc224a5d578cbe45e78e7191539c5dd594156e4aab53aa","target":"graph","created_at":"2026-05-18T00:53:40Z","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 study the asymptotic behaviour of Betti numbers, twisted torsion and other spectral invariants of sequences of locally symmetric spaces. Our main results are uniform versions of the DeGeorge--Wallach Theorem, of a theorem of Delorme and various other limit multiplicity theorems.\n  A basic idea is to adapt the notion of Benjamini--Schramm convergence (BS-convergence), originally introduced for sequences of finite graphs of bounded degree, to sequences of Riemannian manifolds, and analyze the possible limits. We show that BS-convergence of locally symmetric spaces implies convergence, in an a","authors_text":"Ian Biringer, Iddo Samet, Jean Raimbault, Miklos Abert, Nicolas Bergeron, Nikolay Nikolov, Tsachik Gelander","cross_cats":["math.DG","math.GR","math.KT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2012-10-10T15:59:24Z","title":"On the growth of $L^2$-invariants for sequences of lattices in Lie groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1210.2961","kind":"arxiv","version":4},"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":"sha256:944a29b6b3a12354bae682a89839f5225d6bb5942892eeca9defaf812ca0637d","target":"record","created_at":"2026-05-18T00:53:40Z","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":"9b74a552db6cb2dbe13b4f1733523f6be53afded5e2f0e2d22a999313796b9e4","cross_cats_sorted":["math.DG","math.GR","math.KT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RT","submitted_at":"2012-10-10T15:59:24Z","title_canon_sha256":"92b79eb3aa09a053a207304dce6950ca0551fa11859c08a7abbaf736eaa88ce1"},"schema_version":"1.0","source":{"id":"1210.2961","kind":"arxiv","version":4}},"canonical_sha256":"36bb1fca18227b8fd947bc3583631c151946428be1df0ad35d7068e83bfd9c76","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"36bb1fca18227b8fd947bc3583631c151946428be1df0ad35d7068e83bfd9c76","first_computed_at":"2026-05-18T00:53:40.986475Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:53:40.986475Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"RcUAoUjvQGbZZKYmRWT1Xy0Np3ZVBB5uBRlQ166G9zwHYaOxmCFPEASPR7VSfpstriwfKUaDl7WubH9nbWHPDA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:53:40.987029Z","signed_message":"canonical_sha256_bytes"},"source_id":"1210.2961","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:944a29b6b3a12354bae682a89839f5225d6bb5942892eeca9defaf812ca0637d","sha256:b6e8f5b18b70276993fc224a5d578cbe45e78e7191539c5dd594156e4aab53aa"],"state_sha256":"febed645ce1374400fe00666657128ab3bcbb0ff362bbb189a0ce0bc977f1bf3"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yjmXJbVuDyzC5aDl7kzIpDlT/2Zs8bbgW21P9+4LqL4FPrXkiw0CKZD1MYrIQ9TfyI5zUgJTwJaM4GKrwEmwBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T20:37:33.412003Z","bundle_sha256":"0b87ac4e04a51025488e909b9055fff6c45ab1ad6c96961d54ef210d2ad32900"}}