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.
Polynomial effective equidistribution
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abstract
We prove effective equidistribution theorems, with polynomial error rate, for orbits of the unipotent subgroups of $\operatorname{SL}_2(\mathbb R)$ in arithmetic quotients of $\operatorname{SL}_2(\mathbb C)$ and $\operatorname{SL}_2(\mathbb R)\times\operatorname{SL}_2(\mathbb R)$. The proof is based on the use of a Margulis function, tools from incidence geometry, and the spectral gap of the ambient space.
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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.