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We show that the multiplicity of the $k$th eigenvalue of the Laplacian on the Cayley graph of $G$ is bounded by a function of only $c$ and $k$. More specifically, the multiplicity is at most $\\exp((\\log c)(\\log c + \\log k))$.\n  Similarly, if $X$ is a compact, $n$-dimensional Riemannian manifold with non-negative Ricci curvature, then the multiplicity of the $k$t"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"0806.1745","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2008-06-10T20:34:17Z","cross_cats_sorted":["math.DG","math.GR"],"title_canon_sha256":"5c4aa7323620a58c7d2f927676d48057e150e6d86c705ccb203ea22eb7c2af31","abstract_canon_sha256":"2a8a48bd76bd3a64d56a7930945e82fdc895574f7067de71d057f84b82e843cb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-04T15:40:05.802054Z","signature_b64":"fXWlN84V4yW7xvqAWs0/b54kCUrA7MEy4ywaMpw4UOLXss7DBvNBQiITbbuxtYngv4FwYaQyZKlSp6DZujCaDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"719b144a5b03949709008f3d4a737b6400081752f859b45a9c47b5ecf141eda2","last_reissued_at":"2026-07-04T15:40:05.801625Z","signature_status":"signed_v1","first_computed_at":"2026-07-04T15:40:05.801625Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Eigenvalue multiplicity and volume growth","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG","math.GR"],"primary_cat":"math.MG","authors_text":"James R. 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More specifically, the multiplicity is at most $\\exp((\\log c)(\\log c + \\log k))$.\n  Similarly, if $X$ is a compact, $n$-dimensional Riemannian manifold with non-negative Ricci curvature, then the multiplicity of the $k$t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0806.1745","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/0806.1745/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"0806.1745","created_at":"2026-07-04T15:40:05.801689+00:00"},{"alias_kind":"arxiv_version","alias_value":"0806.1745v2","created_at":"2026-07-04T15:40:05.801689+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0806.1745","created_at":"2026-07-04T15:40:05.801689+00:00"},{"alias_kind":"pith_short_12","alias_value":"OGNRISS3AOKJ","created_at":"2026-07-04T15:40:05.801689+00:00"},{"alias_kind":"pith_short_16","alias_value":"OGNRISS3AOKJOCIA","created_at":"2026-07-04T15:40:05.801689+00:00"},{"alias_kind":"pith_short_8","alias_value":"OGNRISS3","created_at":"2026-07-04T15:40:05.801689+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/OGNRISS3AOKJOCIAR46UU433MQ","json":"https://pith.science/pith/OGNRISS3AOKJOCIAR46UU433MQ.json","graph_json":"https://pith.science/api/pith-number/OGNRISS3AOKJOCIAR46UU433MQ/graph.json","events_json":"https://pith.science/api/pith-number/OGNRISS3AOKJOCIAR46UU433MQ/events.json","paper":"https://pith.science/paper/OGNRISS3"},"agent_actions":{"view_html":"https://pith.science/pith/OGNRISS3AOKJOCIAR46UU433MQ","download_json":"https://pith.science/pith/OGNRISS3AOKJOCIAR46UU433MQ.json","view_paper":"https://pith.science/paper/OGNRISS3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0806.1745&json=true","fetch_graph":"https://pith.science/api/pith-number/OGNRISS3AOKJOCIAR46UU433MQ/graph.json","fetch_events":"https://pith.science/api/pith-number/OGNRISS3AOKJOCIAR46UU433MQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/OGNRISS3AOKJOCIAR46UU433MQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/OGNRISS3AOKJOCIAR46UU433MQ/action/storage_attestation","attest_author":"https://pith.science/pith/OGNRISS3AOKJOCIAR46UU433MQ/action/author_attestation","sign_citation":"https://pith.science/pith/OGNRISS3AOKJOCIAR46UU433MQ/action/citation_signature","submit_replication":"https://pith.science/pith/OGNRISS3AOKJOCIAR46UU433MQ/action/replication_record"}},"created_at":"2026-07-04T15:40:05.801689+00:00","updated_at":"2026-07-04T15:40:05.801689+00:00"}