{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:V36OWMK3ENJBR6PWQ3TCM6IWRN","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":"5dce90d765fcc835c3f2264b47ce3342932f7ae828529dd148268c2021fa6047","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-01-20T20:24:02Z","title_canon_sha256":"e1b840f6ea51f619422c1a43e94c7b2018079132e7ea53b7245274101cc78c1f"},"schema_version":"1.0","source":{"id":"1001.3664","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1001.3664","created_at":"2026-05-18T03:55:47Z"},{"alias_kind":"arxiv_version","alias_value":"1001.3664v1","created_at":"2026-05-18T03:55:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1001.3664","created_at":"2026-05-18T03:55:47Z"},{"alias_kind":"pith_short_12","alias_value":"V36OWMK3ENJB","created_at":"2026-05-18T12:26:15Z"},{"alias_kind":"pith_short_16","alias_value":"V36OWMK3ENJBR6PW","created_at":"2026-05-18T12:26:15Z"},{"alias_kind":"pith_short_8","alias_value":"V36OWMK3","created_at":"2026-05-18T12:26:15Z"}],"graph_snapshots":[{"event_id":"sha256:c8910f0ed91c2f07a2492ad3c9557596582857c292482dd5d6b3f8f54254c039","target":"graph","created_at":"2026-05-18T03:55:47Z","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 S be a fixed symmetric finite subset of SL_d(O_K) that generates a Zariski dense subgroup of SL_d(O_K) when we consider it as an algebraic group over Q by restriction of scalars. We prove that the Cayley graphs of SL_d(O_K/I) with respect to the projections of S is an expander family if I ranges over square-free ideals of O_K if d=2 and K is an arbitrary numberfield, or if d=3 and K=Q.","authors_text":"P\\'eter P. Varj\\'u","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-01-20T20:24:02Z","title":"Expansion in $SL_d(O_K/I)$, $I$ square-free"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1001.3664","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":"sha256:285b5a34297678a3ce77f8864f834549623abe6d3319b4dd7fea05bd65ad473c","target":"record","created_at":"2026-05-18T03:55:47Z","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":"5dce90d765fcc835c3f2264b47ce3342932f7ae828529dd148268c2021fa6047","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2010-01-20T20:24:02Z","title_canon_sha256":"e1b840f6ea51f619422c1a43e94c7b2018079132e7ea53b7245274101cc78c1f"},"schema_version":"1.0","source":{"id":"1001.3664","kind":"arxiv","version":1}},"canonical_sha256":"aefceb315b235218f9f686e62679168b65d8726091ac2d15c7b4529c0f5b512e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"aefceb315b235218f9f686e62679168b65d8726091ac2d15c7b4529c0f5b512e","first_computed_at":"2026-05-18T03:55:47.326916Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:55:47.326916Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"1bpHi4QYHOpA3llRuS+BJoAHduBadmNLELNvRThEFuPQzd/kI2V4Vzt+0MYCCdVktOu33MWwX2FVIrfBiLJpDA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:55:47.327360Z","signed_message":"canonical_sha256_bytes"},"source_id":"1001.3664","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:285b5a34297678a3ce77f8864f834549623abe6d3319b4dd7fea05bd65ad473c","sha256:c8910f0ed91c2f07a2492ad3c9557596582857c292482dd5d6b3f8f54254c039"],"state_sha256":"c77a9819f4842308c5d2daf1c72e1b79ec8d4f3a816eea96c22b7a28c919db00"}