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  "body": "\n## Definition\n\nLet $(X, d)$ be a [[metric space|0VKD1TP7F3]] that is geodesic: any two points are joined by a geodesic segment.\nA geodesic segment is an isometric image of a closed interval of $\\R$.\nA geodesic triangle in $X$ consists of three points and three geodesic segments joining them in pairs.\nFor $\\delta \\ge 0$, a geodesic triangle is *$\\delta$-slim* if each of its sides lies in the closed $\\delta$-neighbourhood of the union of the other two sides.\nThe space $X$ is *$\\delta$-hyperbolic* if every geodesic triangle in $X$ is $\\delta$-slim.\n\nLet $G$ be a [[finitely generated group|082DWWXMRJ]] and let $S$ be a finite [[generating set|0RP7VWQ0YN]].\nGive the [[Cayley graph|0GYQK9NDJF]] $\\Gamma(G, S)$ the path metric in which each edge has length $1$.\nThe group $G$ is *hyperbolic* if $\\Gamma(G, S)$ is $\\delta$-hyperbolic for some $\\delta \\ge 0$.\n\n## Remarks\n\nThe definition does not depend on $S$: Cayley graphs for different finite generating sets are [[quasi-isometric|6FKYSH729T]], and among geodesic spaces hyperbolicity is invariant under quasi-isometry, though $\\delta$ may change.\n\nExample: the [[free group|0KTW26C80J]] $F_2$ with a free basis $S$.\nIts Cayley graph is a tree, in which every geodesic triangle is a tripod, so each side lies in the union of the other two and the graph is $0$-hyperbolic.\n\nExample: every finite group is hyperbolic, with $\\delta$ equal to the diameter of its Cayley graph.\n\nNon-example: $\\Z^2$ is not hyperbolic.\nIts Cayley graph is quasi-isometric to the Euclidean plane.\nIn the plane, take the triangle with vertices $(0, 0)$, $(n, 0)$ and $(0, n)$.\nThe point $(n/2, n/2)$ of its hypotenuse is at distance $n/2$ from the other two sides, which is unbounded in $n$.\n\nEquivalent definitions use the Gromov product, or ask that the group be finitely presented with a linear [[Dehn function|Q4HGMRA9JQ]].\nSuch groups are also called word-hyperbolic or Gromov hyperbolic groups.\n",
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