For non-lattice discrete subgroups of higher-rank simple Lie groups, the maximal injectivity radius on balls of radius r grows at least c log log log log r.
Kesten’s theorem for invariant random subgroups.Duke Math
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Applications of Almost Stationarity I: Quantitative Growth of Injectivity Radius and St\"{u}ck-Zimmer Theorem
For non-lattice discrete subgroups of higher-rank simple Lie groups, the maximal injectivity radius on balls of radius r grows at least c log log log log r.