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  "body": "\n## Statement\n\nFor every $n \\ge 1$ there is a constant $\\varepsilon_n \u003e 0$ with the following property.\nLet $X$ be a complete, simply connected [[Riemannian manifold|7CBT4SM83Y]] of dimension $n$ with [[sectional curvature|1X42JAA349]] between $-1$ and $0$.\nLet $\\Gamma$ be a discrete group of [[isometries|02HPP5VCZH]] of $X$, and let $x \\in X$.\nLet $\\Gamma_x$ be the subgroup generated by the elements $\\gamma \\in \\Gamma$ with $d(x, \\gamma x) \u003c \\varepsilon_n$.\nThen $\\Gamma_x$ is virtually nilpotent: it has a [[nilpotent|0Q7MMQHC1P]] subgroup of finite [[index|0DTXBQPA2S]].\n\n## Remarks\n\nThe proof combines a [[Zassenhaus neighbourhood|FZ798EZ6M0]] in a suitable Lie group with a compactness argument.\nThe curvature bounds give uniform control of the isometries that move a point only a little, which is why $\\varepsilon_n$ depends only on $n$.\n\nExample: in the upper half-plane model of $\\mathbb{H}^2$, let $\\Gamma$ be generated by the parabolic isometry $\\gamma(z) = z + 1$.\nAt $x = i y$, the distance $d(x, \\gamma x) = 2 \\operatorname{arsinh}(1/(2y))$ tends to $0$ as $y \\to \\infty$.\nSo for large $y$ the group $\\Gamma_x$ is all of $\\Gamma \\cong \\Z$, which is abelian, as the lemma allows.\n\nThe lower curvature bound matters.\nMultiplying the metric of $\\mathbb{H}^2$ by $1/c$ gives curvature $-c$ and divides all displacements by $\\sqrt{c}$.\nFor $c$ large, the generators of the fundamental group of a closed hyperbolic surface then move a point less than $\\varepsilon_2$.\nThis group is not virtually nilpotent, so the conclusion fails without the bound $-1$.\n\n## Explanations\n\n### L1 human\n\nConsider a group of symmetries of a negatively curved space, such as the hyperbolic plane, that acts without accumulating.\nLook only at the symmetries that move a given point by a very small amount.\nThe lemma says that these symmetries generate a group that is almost commutative, namely virtually nilpotent.\nThe threshold for small depends only on the dimension.\n\n### L2 human\n\nThe main application is the thick-thin decomposition of a hyperbolic manifold $M = \\mathbb{H}^n/\\Gamma$.\nThe thin part, where the injectivity radius is less than $\\varepsilon_n / 2$, consists of tubes around short closed geodesics and of cusps.\nThis structure underlies the study of volumes of hyperbolic manifolds and of their convergence.\nA Lie group form, with lattices in place of $\\Gamma$, is closely related to the [[Kazhdan–Margulis theorem|C34RZMKKA4]].\n",
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