Applications of Almost Stationarity I: Quantitative Growth of Injectivity Radius and Stück-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 $Γ\leq G$, the injectivity radius of points in $G/Γ$ 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 $G/Γ$. More explicitly, we prove that for any $R>0$, one can embed a ball of radius $c\log^{(4)}R$ in $G/Γ$ centered at some point $[g]\in G/Γ$ where $g$ is taken from $G_R$ and for some constant $c=c(G,Γ)$. In particular, we show that for a general discrete subgroup $Γ$, if the injectivity radius growth in $G/Γ$ is slower than $\log^{(4)}$, $Γ$ must be a lattice. Additionally, we give a new, shorter and simpler proof of the Nevo-Stück-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 $Γ\leq G$ is a lattice if and only if there is a probability measure on $G/Γ$ which is sufficiently almost invariant under $G$. More precisely, suppose $Γ\leq G$ is a discrete subgroup for which there exists a probability measure $ν$ on $G/Γ$ for which $W_1^{b}(gν,ν)\leq \eps_0$ for some $\eps_0(Γ)>0$, then $Γ$ is a lattice.