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  "body": "\n## Statement\n\nLet $X$ be a [[proper metric space|0RN765S7DH]] that is geodesic: any two points $x, y$ are joined by a path of length $d(x, y)$.\nLet a group $G$ [[act|0WCY42EV9V]] on $X$ by [[isometries|02HPP5VCZH]].\nAssume that the action is proper: for every compact set $K \\subseteq X$, the set $\\set{g \\in G : gK \\cap K \\neq \\emptyset}$ is finite.\nAssume also that the action is cocompact: there is a compact set $K \\subseteq X$ with $GK = X$.\nThen:\n\n1. $G$ is [[finitely generated|082DWWXMRJ]].\n2. Let $S$ be a finite generating set of $G$ and $x_0 \\in X$. Then the orbit map $g \\mapsto g x_0$ is a [[quasi-isometry|6FKYSH729T]] from $G$, with the [[word metric|VCHJVR9NGH]] of $S$, to $X$.\n\n## Remarks\n\nIdea of the proof.\nChoose a closed ball $B$ around $x_0$ whose translates cover $X$, and let $S$ be the finite set of $g$ with $gB \\cap B \\neq \\emptyset$.\nA geodesic from $x_0$ to $g x_0$ can be cut into short pieces, and consecutive pieces lie in translates of $B$ that meet.\nThis writes $g$ as a word in $S$ of length at most a linear function of $d(x_0, g x_0)$.\nThe other inequality holds because each generator moves $x_0$ by a bounded distance.\n\nExample: $\\Z^n$ acts on $\\R^n$ by translations, properly and cocompactly, so $\\Z^n$ with its standard word metric is quasi-isometric to $\\R^n$.\nExample: the fundamental group of a closed Riemannian manifold acts on the universal cover in this way, so the group and the universal cover are quasi-isometric.\n\nCocompactness is necessary.\nThe group $\\Z$ acts properly on $\\R^2$ by $n \\cdot (a, b) = (a + n, b)$.\nBut the orbit of a point lies on a line, which is not within bounded distance of every point of the plane, so the orbit map is not a quasi-isometry.\n\n## Explanations\n\n### L1 human\n\nSuppose a group moves the points of a nice space around, without piling up, and the moved copies of one bounded region cover the space.\nThen the group and the space have the same large-scale shape.\nSo one can study a group by looking at a space it acts on, and conversely.\nFor instance, the grid $\\Z^2$ and the plane look the same from far away.\n\n### L2 human\n\nThe lemma is the bridge between groups and geometry in [[geometric group theory|2T3218RXTD]].\nIt shows that the quasi-isometry class of a finitely generated group does not depend on the generating set.\nIt also equals the class of any space on which the group acts properly and cocompactly.\nFor instance, the fundamental group of a closed hyperbolic surface is quasi-isometric to the hyperbolic plane, and so it is a [[hyperbolic group|JWJ8YPR7EB]].\nLarge-scale invariants of spaces, such as [[growth|4CMRM3975W]], hyperbolicity or the number of ends, thus become invariants of groups.\n",
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