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We prove that there are constants $C_X,R_X<\\infty$, depending only on $X$, such that every torsion-free lattice $\\Gamma<G$, with $M=\\Gamma\\backslash X$ and global injectivity radius $R=\\operatorname{InjRad}(M)\\ge R_X$, satisfies \\[ \\max\\left\\{d(\\Gamma), \\log\\left|H_1(M;\\mathbb Z)_{\\mathrm{tors}}\\right|\\right\\} \\le C_X\\mathrm{vol}(M)R^{(1-r)/2}(\\log R)^2 \\] where $d(\\Gamma)$ denotes the minim"},"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":"2608.05037","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2026-08-05T16:44:23Z","cross_cats_sorted":["math.DG","math.MG","math.NT"],"title_canon_sha256":"3fd99a3e6ac8bb5be54e5cc9ea61cf9e8bb09052468fead62f9fbe5e19f80ee7","abstract_canon_sha256":"105282c9f5c84fcdde64c52c27333f9299cf4bc5d59c4a368a32c7b255efbaa0"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-08-06T01:48:05.424875Z","signature_b64":"xdCq6l2/K96lc7J16MMSNxQ8oczVGAYQ6Za9zg45cWfuwRhKjhsR35O5oP66v1jewj7d44P9kAGperPqywIRDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"935799411b372f2127783200f43ce742b7f7ea4ab3a8e9915342d2f1afdc0983","last_reissued_at":"2026-08-06T01:48:05.423496Z","signature_status":"signed_v1","first_computed_at":"2026-08-06T01:48:05.423496Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Sparse Random Covers and Growth of Torsion in First Homology","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG","math.MG","math.NT"],"primary_cat":"math.GR","authors_text":"Raz Slutsky","submitted_at":"2026-08-05T16:44:23Z","abstract_excerpt":"We construct random open covers of higher-rank locally symmetric spaces using a construction we call scaffolded Poisson processes. 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