{
  "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 $Γ\\leq G$, 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-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 $Γ\\leq G$ 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.01382",
  "author_total": 4,
  "authors": [
    {
      "name": "Ilya Gekhtman"
    },
    {
      "name": "Simon Machado"
    },
    {
      "name": "Omri Solan"
    },
    {
      "name": "Yuval Yifrach"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2gvhhv3pffm7k.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.01382",
  "fulltext_url": null,
  "id": "2gvhhv3pffm7k",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.01382",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.DS"
    ],
    [
      "openalex",
      "2610|Advanced Algebra and Geometry"
    ]
  ],
  "openalex": "W7172424402",
  "pdf_url": "https://arxiv.org/pdf/2608.01382",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2gvhhv3pffm7k.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2608.01382",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.01382"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.DS",
      "subfield": "dynamical-systems"
    }
  ],
  "title": "Applications of Almost Stationarity I: Quantitative Growth of Injectivity Radius and Stück-Zimmer Theorem",
  "title_text": "Applications of Almost Stationarity I: Quantitative Growth of Injectivity Radius and Stück-Zimmer Theorem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/2gvhhv3pffm7k",
  "venue": "arXiv",
  "versions": []
}