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It follows that under appropriate conditions the crossed product $C^*$-algebra $\\cl A$ associated with the action of $\\Gamma$ on the boundary of $\\cl T_1\\times \\cl T_2$ satisfies $\\rank K_0(\\cl A) = 2\\cdot\\rank H_2(\\Gamma, \\bb Z)$."},"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":"math/0511447","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.KT","submitted_at":"2005-11-17T17:03:14Z","cross_cats_sorted":["math.OA"],"title_canon_sha256":"765257e932040cb0a7de1b8d1a664e55cf0962c67abdf7b13672b3363ece943f","abstract_canon_sha256":"2a619f04cba4e4d6ecf2ecb439d0ce9067ef0459b574083d0ca8e8e20657259d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:32:45.905125Z","signature_b64":"grAQbtwHmZus23MtDYSFQZ4MrIUIZLpdbEkHBjRYiqDEhbveP4Wdaljt9lvQo20ar7xLis5kSdJ2eD2w0945Cg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"69be7699c109dd774915fed1f0c8cde9a09d35d54ca971abf59e14b741887ce5","last_reissued_at":"2026-05-18T03:32:45.904385Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:32:45.904385Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Tiling systems and homology of lattices in tree products","license":"","headline":"","cross_cats":["math.OA"],"primary_cat":"math.KT","authors_text":"Guyan Robertson","submitted_at":"2005-11-17T17:03:14Z","abstract_excerpt":"Let $\\Gamma$ be a torsion free cocompact lattice in $\\aut(\\cl T_1)\\times\\aut(\\cl T_2)$, where $\\cl T_1$, $\\cl T_2$ are trees whose vertices all have degree at least three. 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