{"paper":{"title":"Applications of Almost Stationarity I: Quantitative Growth of Injectivity Radius and St\\\"{u}ck-Zimmer Theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Ilya Gekhtman, Omri Solan, Simon Machado, Yuval Yifrach","submitted_at":"2026-08-02T17:04:23Z","abstract_excerpt":"Fraczyk and Gelander proved in \\cite{FG} that for any simple Lie group $G$ of high rank and for every non-lattice discrete subgroup $\\Gamma\\leq G$, the injectivity radius of points in $G/\\Gamma$ is unbounded, resolving a conjecture of Margulis.\n  In this work we obtain an explicit lower bound on the growth rate of the maximal injectivity radius of points taken from growing balls in $G/\\Gamma$.\n  More explicitly, we prove that for any $R>0$, one can embed a ball of radius $c\\log^{(4)}R$ in $G/\\Gamma$ centered at some point $[g]\\in G/\\Gamma$ where $g$ is taken from $G_R$ and for some constant $c"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.01382","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/2608.01382/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"}