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We prove a zero-one law for the probability that $G(q)$ is generated by a random $r$-tuple in $X(q) = X\\cap G(q)^r$: the limit of this probability as $q$ increases (through values of $q$ for which $X$ is stable under the Frobenius morphism defining $G(q)$) is either 1 or 0. Indeed, to ensure that this limit is 1, one only needs $G(q)$ to be"},"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":"1810.01737","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2018-10-03T13:47:04Z","cross_cats_sorted":[],"title_canon_sha256":"e4de22b5d893c35650fd67a0261d83c37f5d78a90686a4b1e8e4a992cb99f82a","abstract_canon_sha256":"719ae4ea50187b0d2375b1603ef12e74e22e5b99080e8712ab91722e3752e364"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:04:11.128215Z","signature_b64":"yAGw/PkUI48jOIUvmOE6sUN1+vYAtg7Di1QJ5EDhK8MaUWyZo6uax5kxdGrF2ff65PNjpRD3/SKHaxisOr4OCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9c64b495bc43c7ea58a76ead7e89f6c8a8106d08f36c51aa311dd3576a9b70e5","last_reissued_at":"2026-05-18T00:04:11.127628Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:04:11.127628Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Zero-one generation laws for finite simple groups","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Aner Shalev, Frank L\\\"ubeck, Martin W. Liebeck, Robert M. Guralnick","submitted_at":"2018-10-03T13:47:04Z","abstract_excerpt":"Let $G$ be a simple algebraic group over the algebraic closure of $GF(p)$ ($p$ prime), and let $G(q)$ denote a corresponding finite group of Lie type over $GF(q)$, where $q$ is a power of $p$. Let $X$ be an irreducible subvariety of $G^r$ for some $r\\ge 2$. We prove a zero-one law for the probability that $G(q)$ is generated by a random $r$-tuple in $X(q) = X\\cap G(q)^r$: the limit of this probability as $q$ increases (through values of $q$ for which $X$ is stable under the Frobenius morphism defining $G(q)$) is either 1 or 0. 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