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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"},"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":"1104.4416","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.KT","submitted_at":"2011-04-22T10:21:43Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"9aed7049dc23dde72c088589b6cc5f86c6bd8ec0b4ae58b5753b6cd682addc49","abstract_canon_sha256":"fe37fa7979124fcb16a1c07d2798d32ae1ea63c4e4abbfa2c6e6c9e9b2f4f794"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:32:57.343542Z","signature_b64":"0wUE6UnHkrYqjikbhTof0P5NAIjEqdHawhdRrnvsNavgh6fotcurST+YkViTytLr77l/cgxXfssfmyzw7UiaDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9a232177742c4d61aa226bac1349f80f6f65478ff4a3539a8f402d0fc7d7911f","last_reissued_at":"2026-05-18T03:32:57.342691Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:32:57.342691Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the K-theory of boundary $C^*$-algebras of $\\widetilde A_2$ groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.KT","authors_text":"Guyan Robertson, Oliver King","submitted_at":"2011-04-22T10:21:43Z","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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1104.4416","kind":"arxiv","version":1},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1104.4416","created_at":"2026-05-18T03:32:57.342842+00:00"},{"alias_kind":"arxiv_version","alias_value":"1104.4416v1","created_at":"2026-05-18T03:32:57.342842+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1104.4416","created_at":"2026-05-18T03:32:57.342842+00:00"},{"alias_kind":"pith_short_12","alias_value":"TIRSC53UFRGW","created_at":"2026-05-18T12:26:42.757692+00:00"},{"alias_kind":"pith_short_16","alias_value":"TIRSC53UFRGWDKRC","created_at":"2026-05-18T12:26:42.757692+00:00"},{"alias_kind":"pith_short_8","alias_value":"TIRSC53U","created_at":"2026-05-18T12:26:42.757692+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/TIRSC53UFRGWDKRCNOWBGSPYB5","json":"https://pith.science/pith/TIRSC53UFRGWDKRCNOWBGSPYB5.json","graph_json":"https://pith.science/api/pith-number/TIRSC53UFRGWDKRCNOWBGSPYB5/graph.json","events_json":"https://pith.science/api/pith-number/TIRSC53UFRGWDKRCNOWBGSPYB5/events.json","paper":"https://pith.science/paper/TIRSC53U"},"agent_actions":{"view_html":"https://pith.science/pith/TIRSC53UFRGWDKRCNOWBGSPYB5","download_json":"https://pith.science/pith/TIRSC53UFRGWDKRCNOWBGSPYB5.json","view_paper":"https://pith.science/paper/TIRSC53U","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1104.4416&json=true","fetch_graph":"https://pith.science/api/pith-number/TIRSC53UFRGWDKRCNOWBGSPYB5/graph.json","fetch_events":"https://pith.science/api/pith-number/TIRSC53UFRGWDKRCNOWBGSPYB5/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TIRSC53UFRGWDKRCNOWBGSPYB5/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TIRSC53UFRGWDKRCNOWBGSPYB5/action/storage_attestation","attest_author":"https://pith.science/pith/TIRSC53UFRGWDKRCNOWBGSPYB5/action/author_attestation","sign_citation":"https://pith.science/pith/TIRSC53UFRGWDKRCNOWBGSPYB5/action/citation_signature","submit_replication":"https://pith.science/pith/TIRSC53UFRGWDKRCNOWBGSPYB5/action/replication_record"}},"created_at":"2026-05-18T03:32:57.342842+00:00","updated_at":"2026-05-18T03:32:57.342842+00:00"}