Applications of Almost Stationarity I: Quantitative Growth of Injectivity Radius and Stück-Zimmer Theorem
Abstract
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≤ 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 clog(4)R in G/Γ centered at some point [g]∈ G/Γ where g is taken from GR 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-Stuck-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≤ 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≤ G is a discrete subgroup for which there exists a probability measure ν on G/Γ for which W1b(gν,ν)≤ eps0 for some eps0(Γ)>0, then Γ is a lattice.
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BibTeX
@misc{gekhtman2026applications,
title = {{Applications of Almost Stationarity I: Quantitative Growth of Injectivity Radius and Stück-Zimmer Theorem}},
author = {Ilya Gekhtman and Simon Machado and Omri Solan and Yuval Yifrach},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.01382},
eprint = {2608.01382},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.01382},
}
RIS
TY - UNPB
TI - Applications of Almost Stationarity I: Quantitative Growth of Injectivity Radius and Stück-Zimmer Theorem
AU - Ilya Gekhtman
AU - Simon Machado
AU - Omri Solan
AU - Yuval Yifrach
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2608.01382
UR - https://arxiv.org/abs/2608.01382
AB - 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≤ 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 clog^(4)R in G/Γ centered at some point [g]∈ 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-Stuck-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≤ 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≤ G is a discrete subgroup for which there exists a probability measure ν on G/Γ for which W_1^b(gν,ν)≤ eps_0 for some eps_0(Γ)>0, then Γ is a lattice.
ER -