{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:FRLUUIVGDKMZB3NFMQQE5HWHDU","short_pith_number":"pith:FRLUUIVG","canonical_record":{"source":{"id":"1102.0031","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2011-01-31T23:07:29Z","cross_cats_sorted":["math.FA","math.RT"],"title_canon_sha256":"68140ed26620d9efe96f35863cc3133b1485a3c31fb00ddb5f7fec0eb8e98d5e","abstract_canon_sha256":"33cc357a33941ee949f2180a5e12a24017cf578b9c391e2c3631c355f644eb10"},"schema_version":"1.0"},"canonical_sha256":"2c574a22a61a9990eda564204e9ec71d164effda870d4fa530568bc30b4e4439","source":{"kind":"arxiv","id":"1102.0031","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1102.0031","created_at":"2026-05-18T02:56:44Z"},{"alias_kind":"arxiv_version","alias_value":"1102.0031v2","created_at":"2026-05-18T02:56:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1102.0031","created_at":"2026-05-18T02:56:44Z"},{"alias_kind":"pith_short_12","alias_value":"FRLUUIVGDKMZ","created_at":"2026-05-18T12:26:28Z"},{"alias_kind":"pith_short_16","alias_value":"FRLUUIVGDKMZB3NF","created_at":"2026-05-18T12:26:28Z"},{"alias_kind":"pith_short_8","alias_value":"FRLUUIVG","created_at":"2026-05-18T12:26:28Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:FRLUUIVGDKMZB3NFMQQE5HWHDU","target":"record","payload":{"canonical_record":{"source":{"id":"1102.0031","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2011-01-31T23:07:29Z","cross_cats_sorted":["math.FA","math.RT"],"title_canon_sha256":"68140ed26620d9efe96f35863cc3133b1485a3c31fb00ddb5f7fec0eb8e98d5e","abstract_canon_sha256":"33cc357a33941ee949f2180a5e12a24017cf578b9c391e2c3631c355f644eb10"},"schema_version":"1.0"},"canonical_sha256":"2c574a22a61a9990eda564204e9ec71d164effda870d4fa530568bc30b4e4439","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:56:44.625277Z","signature_b64":"IG+2bvhRIV0yvzoAerjzPl3B8iy9WNczbOUlVeGrg/UVTRq/P3zSk+kTVN8CTWvXWKsHPzaLIGpJwHOEZDu4Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2c574a22a61a9990eda564204e9ec71d164effda870d4fa530568bc30b4e4439","last_reissued_at":"2026-05-18T02:56:44.624753Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:56:44.624753Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1102.0031","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:56:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wvqDXrULZ/RR0gqjtqWdFvn8m2mwJmeeTQqgXALvkBTjVfW9llT6KtqstAtuhTS4Sj6kuI+GkYjaTMlojCoHAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T20:45:55.994171Z"},"content_sha256":"8ed323fef7ff0dbf8ee81d4c4f2eab434b1fbd7007e5e3e307ec0d37cd7ac1f5","schema_version":"1.0","event_id":"sha256:8ed323fef7ff0dbf8ee81d4c4f2eab434b1fbd7007e5e3e307ec0d37cd7ac1f5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:FRLUUIVGDKMZB3NFMQQE5HWHDU","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Property (T) for groups graded by root systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA","math.RT"],"primary_cat":"math.GR","authors_text":"Andrei Jaikin-Zapirain, Martin Kassabov, Mikhail Ershov","submitted_at":"2011-01-31T23:07:29Z","abstract_excerpt":"We introduce and study the class of groups graded by root systems. We prove that if {\\Phi} is an irreducible classical root system of rank at least 2 and G is a group graded by {\\Phi}, then under certain natural conditions on the grading, the union of the root subgroups is a Kazhdan subset of G. As the main application of this theorem we prove that for any reduced irreducible classical root system {\\Phi} of rank at least 2 and a finitely generated commutative ring R with 1, the Steinberg group St_{\\Phi}(R) and the elementary Chevalley group E_{\\Phi}(R) have property (T). We also show that ther"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.0031","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":""},"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":"2026-05-18T02:56:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"V/DlZJuskMrYdiya2A/k1ADeVllbJmLVr6tLxl91mURfSao2did2IzlNz1olSOLUI4K7OiXJvMMasSVrl86/Dw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T20:45:55.994533Z"},"content_sha256":"7e594504447d231154366e4faf5167d23345f93be016adc0cd01389e27cd3ac4","schema_version":"1.0","event_id":"sha256:7e594504447d231154366e4faf5167d23345f93be016adc0cd01389e27cd3ac4"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FRLUUIVGDKMZB3NFMQQE5HWHDU/bundle.json","state_url":"https://pith.science/pith/FRLUUIVGDKMZB3NFMQQE5HWHDU/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FRLUUIVGDKMZB3NFMQQE5HWHDU/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":"2026-06-01T20:45:55Z","links":{"resolver":"https://pith.science/pith/FRLUUIVGDKMZB3NFMQQE5HWHDU","bundle":"https://pith.science/pith/FRLUUIVGDKMZB3NFMQQE5HWHDU/bundle.json","state":"https://pith.science/pith/FRLUUIVGDKMZB3NFMQQE5HWHDU/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FRLUUIVGDKMZB3NFMQQE5HWHDU/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:FRLUUIVGDKMZB3NFMQQE5HWHDU","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":"33cc357a33941ee949f2180a5e12a24017cf578b9c391e2c3631c355f644eb10","cross_cats_sorted":["math.FA","math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2011-01-31T23:07:29Z","title_canon_sha256":"68140ed26620d9efe96f35863cc3133b1485a3c31fb00ddb5f7fec0eb8e98d5e"},"schema_version":"1.0","source":{"id":"1102.0031","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1102.0031","created_at":"2026-05-18T02:56:44Z"},{"alias_kind":"arxiv_version","alias_value":"1102.0031v2","created_at":"2026-05-18T02:56:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1102.0031","created_at":"2026-05-18T02:56:44Z"},{"alias_kind":"pith_short_12","alias_value":"FRLUUIVGDKMZ","created_at":"2026-05-18T12:26:28Z"},{"alias_kind":"pith_short_16","alias_value":"FRLUUIVGDKMZB3NF","created_at":"2026-05-18T12:26:28Z"},{"alias_kind":"pith_short_8","alias_value":"FRLUUIVG","created_at":"2026-05-18T12:26:28Z"}],"graph_snapshots":[{"event_id":"sha256:7e594504447d231154366e4faf5167d23345f93be016adc0cd01389e27cd3ac4","target":"graph","created_at":"2026-05-18T02:56:44Z","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 introduce and study the class of groups graded by root systems. We prove that if {\\Phi} is an irreducible classical root system of rank at least 2 and G is a group graded by {\\Phi}, then under certain natural conditions on the grading, the union of the root subgroups is a Kazhdan subset of G. As the main application of this theorem we prove that for any reduced irreducible classical root system {\\Phi} of rank at least 2 and a finitely generated commutative ring R with 1, the Steinberg group St_{\\Phi}(R) and the elementary Chevalley group E_{\\Phi}(R) have property (T). We also show that ther","authors_text":"Andrei Jaikin-Zapirain, Martin Kassabov, Mikhail Ershov","cross_cats":["math.FA","math.RT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2011-01-31T23:07:29Z","title":"Property (T) for groups graded by root systems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.0031","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":"sha256:8ed323fef7ff0dbf8ee81d4c4f2eab434b1fbd7007e5e3e307ec0d37cd7ac1f5","target":"record","created_at":"2026-05-18T02:56:44Z","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":"33cc357a33941ee949f2180a5e12a24017cf578b9c391e2c3631c355f644eb10","cross_cats_sorted":["math.FA","math.RT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2011-01-31T23:07:29Z","title_canon_sha256":"68140ed26620d9efe96f35863cc3133b1485a3c31fb00ddb5f7fec0eb8e98d5e"},"schema_version":"1.0","source":{"id":"1102.0031","kind":"arxiv","version":2}},"canonical_sha256":"2c574a22a61a9990eda564204e9ec71d164effda870d4fa530568bc30b4e4439","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2c574a22a61a9990eda564204e9ec71d164effda870d4fa530568bc30b4e4439","first_computed_at":"2026-05-18T02:56:44.624753Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:56:44.624753Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"IG+2bvhRIV0yvzoAerjzPl3B8iy9WNczbOUlVeGrg/UVTRq/P3zSk+kTVN8CTWvXWKsHPzaLIGpJwHOEZDu4Bg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:56:44.625277Z","signed_message":"canonical_sha256_bytes"},"source_id":"1102.0031","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8ed323fef7ff0dbf8ee81d4c4f2eab434b1fbd7007e5e3e307ec0d37cd7ac1f5","sha256:7e594504447d231154366e4faf5167d23345f93be016adc0cd01389e27cd3ac4"],"state_sha256":"aaa93f3866535f3ac67fe7f604e62cc0659a8571def45f82399f9a13dc6c7ff9"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PhAQeFvNCdAIKB9e5LtPEP+eoiwzMcLoirwxc6yo2ItFO1zEN/cqBYRNhD6ePQ0FfFAT6/bO/DR1tYpEXhgLDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T20:45:55.996500Z","bundle_sha256":"d5f006b59b5964a1fca73a5379b0b696020657b50c03f44380d6d19303e40c9e"}}