Applications of Almost Stationarity I: Quantitative Growth of Injectivity Radius and St\"{u}ck-Zimmer Theorem
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. 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 $G/\Gamma$. 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 $c=c(G,\Gamma)$. In particular, we show that for a general discrete subgroup $\Gamma$, if the injectivity radius growth in $G/\Gamma$ is slower than $\log^{(4)}$, $\Gamma$ must be a lattice. Additionally, we give a new, shorter and simpler proof of the Nevo-St\"{u}ck-Zimmer Theorem, saying that every action of a high rank simple group with property $(T)$ is either essentially free or essentially transitive. The results in this paper are obtained using the almost structure of measures from the accompanying paper, together with additional geometric considerations. As a step in the proof, we develop the following characterization for lattices. A discrete subgroup $\Gamma\leq G$ is a lattice if and only if there is a probability measure on $G/\Gamma$ which is sufficiently almost invariant under $G$. More precisely, suppose $\Gamma\leq G$ is a discrete subgroup for which there exists a probability measure $\nu$ on $G/\Gamma$ for which $W_1^{b}(g\nu,\nu)\leq \eps_0$ for some $\eps_0(\Gamma)>0$, then $\Gamma$ is a lattice.