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.
On the convergence of arithmetic orbifolds
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We discuss the geometry of some arithmetic orbifolds locally isometric to a product of real hyperbolic spaces of dimension two and three, and prove that certain sequences of non-uniform orbifolds are convergent to this space in a geometric ("Benjamini--Schramm") sense for hyperbolic three--space and a product of hyperbolic planes. We also deal with arbitrary sequences of maximal arithmetic three--dimensional hyperbolic lattices defined over a quadratic or cubic field. A motivating application is the study of Betti numbers of Bianchi groups.
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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.