{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:Q2PRQ4BJD7AGKZZKOERSIL64YS","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":"65994ef5e1e9d7402d30d5b66173fe96d29d373ec7b1b083412c6247b5a786a5","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2013-07-01T14:31:12Z","title_canon_sha256":"699a181cfb4791180d938e4df18097226693e4dac8829f1dfef99a8bdf241614"},"schema_version":"1.0","source":{"id":"1307.0379","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1307.0379","created_at":"2026-05-18T03:19:33Z"},{"alias_kind":"arxiv_version","alias_value":"1307.0379v1","created_at":"2026-05-18T03:19:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1307.0379","created_at":"2026-05-18T03:19:33Z"},{"alias_kind":"pith_short_12","alias_value":"Q2PRQ4BJD7AG","created_at":"2026-05-18T12:27:57Z"},{"alias_kind":"pith_short_16","alias_value":"Q2PRQ4BJD7AGKZZK","created_at":"2026-05-18T12:27:57Z"},{"alias_kind":"pith_short_8","alias_value":"Q2PRQ4BJ","created_at":"2026-05-18T12:27:57Z"}],"graph_snapshots":[{"event_id":"sha256:ac4acc0578c8d80a6ac5c84fb83546e6f57308913a4ea165e702b064f474d1af","target":"graph","created_at":"2026-05-18T03:19:33Z","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 compute $L^2$-Betti numbers of postliminal, locally compact, unimodular groups in terms of ordinary dimensions of reduced cohomology with coefficients in irreducible unitary representations and the Plancherel measure. This allows us to compute the $L^2$-Betti numbers for semi-simple Lie groups with finite center, simple algebraic groups over local fields, and automorphism groups of locally finite trees acting transitively on the boundary.","authors_text":"Alain Valette, Henrik Densing Petersen","cross_cats":["math.FA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2013-07-01T14:31:12Z","title":"L^2-Betti numbers and Plancherel measure"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.0379","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":"sha256:44bd97d18caccd0f895980e6d6b84539496acb5b7f15edc104988f6cb712b978","target":"record","created_at":"2026-05-18T03:19:33Z","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":"65994ef5e1e9d7402d30d5b66173fe96d29d373ec7b1b083412c6247b5a786a5","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2013-07-01T14:31:12Z","title_canon_sha256":"699a181cfb4791180d938e4df18097226693e4dac8829f1dfef99a8bdf241614"},"schema_version":"1.0","source":{"id":"1307.0379","kind":"arxiv","version":1}},"canonical_sha256":"869f1870291fc065672a7123242fdcc489ea4cc1ad9d16bb9bd6d0e264f93257","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"869f1870291fc065672a7123242fdcc489ea4cc1ad9d16bb9bd6d0e264f93257","first_computed_at":"2026-05-18T03:19:33.540260Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:19:33.540260Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"in2uYOhSl37WtAqA4uf8dHpUXq8a9H9EaWVtUnpWK9XWlkgdwOsjjZB4MQ8+M4h+DeMsP3TSI3nbr+YEyhmIBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:19:33.540873Z","signed_message":"canonical_sha256_bytes"},"source_id":"1307.0379","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:44bd97d18caccd0f895980e6d6b84539496acb5b7f15edc104988f6cb712b978","sha256:ac4acc0578c8d80a6ac5c84fb83546e6f57308913a4ea165e702b064f474d1af"],"state_sha256":"ea62e41419b9782bfb13a5c07f4ac4b8547004e2ab9efe6ad28e170ef17d717a"}