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.
A view on Invariant Random Subgroups and Lattices
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abstract
For more than half a century lattices in Lie groups played an important role in geometry, number theory and group theory. Recently the notion of Invariant Random Subgroups (IRS) emerged as a natural generalization of lattices. It is thus intriguing to extend results from the theory of lattices to the context of IRS, and to study lattices by analyzing the compact space of all IRS of a given group. This article focuses on the interplay between lattices and IRS, mainly in the classical case of semisimple analytic groups over local fields.
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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.