{"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. Let $X=G/K$ be a symmetric space of noncompact type and real rank $r\\ge2$. 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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.05037","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2608.05037/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}