{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:TIRSC53UFRGWDKRCNOWBGSPYB5","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":"fe37fa7979124fcb16a1c07d2798d32ae1ea63c4e4abbfa2c6e6c9e9b2f4f794","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.KT","submitted_at":"2011-04-22T10:21:43Z","title_canon_sha256":"9aed7049dc23dde72c088589b6cc5f86c6bd8ec0b4ae58b5753b6cd682addc49"},"schema_version":"1.0","source":{"id":"1104.4416","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1104.4416","created_at":"2026-05-18T03:32:57Z"},{"alias_kind":"arxiv_version","alias_value":"1104.4416v1","created_at":"2026-05-18T03:32:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1104.4416","created_at":"2026-05-18T03:32:57Z"},{"alias_kind":"pith_short_12","alias_value":"TIRSC53UFRGW","created_at":"2026-05-18T12:26:42Z"},{"alias_kind":"pith_short_16","alias_value":"TIRSC53UFRGWDKRC","created_at":"2026-05-18T12:26:42Z"},{"alias_kind":"pith_short_8","alias_value":"TIRSC53U","created_at":"2026-05-18T12:26:42Z"}],"graph_snapshots":[{"event_id":"sha256:d43449c78d990f0373fe23c1930f82717bde45707be0bcce1bec8d78d6af6d8a","target":"graph","created_at":"2026-05-18T03:32:57Z","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 $\\Gamma$ be an $\\widetilde A_2$ subgroup of $\\PGL_3(\\mathbb K)$, where $\\mathbb K$ is a local field with residue field of order $q$. The module of coinvariants $C(\\mathbb P^2_{\\mathbb K},\\mathbb Z)_{\\Gamma}$ is shown to be finite, where $\\mathbb P^2_{\\mathbb K}$ is the projective plane over $\\mathbb K$. If the group $\\Gamma$ is of Tits type and if $q \\not\\equiv 1 \\pmod {3}$ then the exact value of the order of the class $[I]_{K_0}$ in the K-theory of the (full) crossed product $C^*$-algebra $C(\\Omega)\\rtimes\\Gamma$ is determined, where $\\Omega$ is the Furstenberg boundary of $\\PGL_3(\\mathb","authors_text":"Guyan Robertson, Oliver King","cross_cats":["math.GR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.KT","submitted_at":"2011-04-22T10:21:43Z","title":"On the K-theory of boundary $C^*$-algebras of $\\widetilde A_2$ groups"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1104.4416","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:a07c64b6745b17953cc2da8eec06772000cc6264b69918a0501c297668a6f634","target":"record","created_at":"2026-05-18T03:32:57Z","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":"fe37fa7979124fcb16a1c07d2798d32ae1ea63c4e4abbfa2c6e6c9e9b2f4f794","cross_cats_sorted":["math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.KT","submitted_at":"2011-04-22T10:21:43Z","title_canon_sha256":"9aed7049dc23dde72c088589b6cc5f86c6bd8ec0b4ae58b5753b6cd682addc49"},"schema_version":"1.0","source":{"id":"1104.4416","kind":"arxiv","version":1}},"canonical_sha256":"9a232177742c4d61aa226bac1349f80f6f65478ff4a3539a8f402d0fc7d7911f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9a232177742c4d61aa226bac1349f80f6f65478ff4a3539a8f402d0fc7d7911f","first_computed_at":"2026-05-18T03:32:57.342691Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:32:57.342691Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"0wUE6UnHkrYqjikbhTof0P5NAIjEqdHawhdRrnvsNavgh6fotcurST+YkViTytLr77l/cgxXfssfmyzw7UiaDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:32:57.343542Z","signed_message":"canonical_sha256_bytes"},"source_id":"1104.4416","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a07c64b6745b17953cc2da8eec06772000cc6264b69918a0501c297668a6f634","sha256:d43449c78d990f0373fe23c1930f82717bde45707be0bcce1bec8d78d6af6d8a"],"state_sha256":"dec170197e6a1871e00cd7ffb85099802857792bf5b273cbec7f1bc3df30eb03"}