{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:E4NVGV5RMFB2K3BDVLJ2GYIRAB","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":"6919458d32a8381325cb0ca2ad3fce2a4f5a296a2568b8b1e85ccc6bc8f726b5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-02-01T07:06:09Z","title_canon_sha256":"f0d615a1dbd347cbd343e10c4f5e2d94d9f4858cf64628f85ffe351212c85d7d"},"schema_version":"1.0","source":{"id":"1602.00409","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1602.00409","created_at":"2026-05-18T00:23:57Z"},{"alias_kind":"arxiv_version","alias_value":"1602.00409v2","created_at":"2026-05-18T00:23:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1602.00409","created_at":"2026-05-18T00:23:57Z"},{"alias_kind":"pith_short_12","alias_value":"E4NVGV5RMFB2","created_at":"2026-05-18T12:30:12Z"},{"alias_kind":"pith_short_16","alias_value":"E4NVGV5RMFB2K3BD","created_at":"2026-05-18T12:30:12Z"},{"alias_kind":"pith_short_8","alias_value":"E4NVGV5R","created_at":"2026-05-18T12:30:12Z"}],"graph_snapshots":[{"event_id":"sha256:7c4b424472cceefd138eb95433f975b9661d5b59408d377dc31459bb9586f88f","target":"graph","created_at":"2026-05-18T00:23:57Z","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":"Let $\\Omega$ be a finite symmetric subset of GL$_n(\\mathbb{Z}[1/q_0])$, and $\\Gamma:=\\langle \\Omega \\rangle$. Then the family of Cayley graphs $\\{{\\rm Cay}(\\pi_m(\\Gamma),\\pi_m(\\Omega))\\}_m$ is a family of expanders as $m$ ranges over fixed powers of square-free integers and powers of primes that are coprime to $q_0$ if and only if the connected component of the Zariski-closure of $\\Gamma$ is perfect. Some of the immediate applications, e.g. orbit equivalence rigidity, {\\em largeness} of certain $\\ell$-adic Galois representations, are also discussed.","authors_text":"Alireza Salehi Golsefidy","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-02-01T07:06:09Z","title":"Super-approximation, II: the p-adic and bounded power of square-free integers cases"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.00409","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":"sha256:71da8630665556df881096db2912a22b48f6757cc589b8d269bd73201ab5861e","target":"record","created_at":"2026-05-18T00:23:57Z","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":"6919458d32a8381325cb0ca2ad3fce2a4f5a296a2568b8b1e85ccc6bc8f726b5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2016-02-01T07:06:09Z","title_canon_sha256":"f0d615a1dbd347cbd343e10c4f5e2d94d9f4858cf64628f85ffe351212c85d7d"},"schema_version":"1.0","source":{"id":"1602.00409","kind":"arxiv","version":2}},"canonical_sha256":"271b5357b16143a56c23aad3a3611100702d9c7da41ef9cdb34da447fccc3755","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"271b5357b16143a56c23aad3a3611100702d9c7da41ef9cdb34da447fccc3755","first_computed_at":"2026-05-18T00:23:57.286735Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:23:57.286735Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"LZtLA/877FbsXXkSAzlGPT9hxsQUpOQTcAW/Ouj3S0Yu360FL01+aFutk/GDzjYpDfK7OlT89JCITw3Wco1cBg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:23:57.287305Z","signed_message":"canonical_sha256_bytes"},"source_id":"1602.00409","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:71da8630665556df881096db2912a22b48f6757cc589b8d269bd73201ab5861e","sha256:7c4b424472cceefd138eb95433f975b9661d5b59408d377dc31459bb9586f88f"],"state_sha256":"a5fd938466c7d6a1638c87454c7a3bcc20aa2a473ccd183d4c466fd4ea19ebba"}