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The crossed product $C^*$-algebra ${\\mathcal A}(\\Gamma)=C(\\Omega) \\rtimes \\Gamma$ depends only on $\\Gamma$ and is classified by its K-theory. This article shows how to compute the K-theory of ${\\mathcal A}(\\Gamma)$ and of the larger class of rank two Cuntz-Krieger algebras."},"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/0011191","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.OA","submitted_at":"2000-11-23T03:35:52Z","cross_cats_sorted":["math.KT"],"title_canon_sha256":"db5ffa929d806d57b1375bc2dbc84210498896b4a89b02ee8df5bfa8f92e5067","abstract_canon_sha256":"cc262e678512b7a825a387a1d42c7b496873559d45a5b9797fa4edd1d96afd77"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:32:45.991822Z","signature_b64":"E8GnXUOI+Jp+mpD3EMhzB3TvaukC9CcSQw5wBj9rwbUg7HwHoWuNhm8yHmSkXq9u6bNOwppfA0fjhY/orD+yCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"377ce60678b810a546691bba3a35e8696cb59c3459ca55531d2a59dc4ae27b6a","last_reissued_at":"2026-05-18T03:32:45.991134Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:32:45.991134Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Asymptotic K-theory for groups acting on $\\tildeA_2$ buildings","license":"","headline":"","cross_cats":["math.KT"],"primary_cat":"math.OA","authors_text":"Guyan Robertson, Tim Steger","submitted_at":"2000-11-23T03:35:52Z","abstract_excerpt":"Let $\\Gamma$ be a torsion free lattice in $G=\\PGL(3,{{\\mathbb F}})$ where ${{\\mathbb F}}$ is a nonarchimedean local field. 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