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  "body": "\n## Definition\n\nLet $\\mathrm{PSL}_2(\\R) = \\mathrm{SL}_2(\\R) / \\set{\\pm I}$, where [[SL(2,R)|Q0GBCB9RA4]] has the topology of a subset of $\\R^4$ and $\\mathrm{PSL}_2(\\R)$ has the quotient topology.\nA *Fuchsian group* is a [[discrete subgroup|5M7D1BSHVZ]] of $\\mathrm{PSL}_2(\\R)$.\n\n## Remarks\n\nThe group $\\mathrm{PSL}_2(\\R)$ acts on the [[upper half-plane|X6XGGH5FPW]] $\\mathbb H$ by [[Möbius transformations|0SG74WBFA2]] $z \\mapsto (az + b)/(cz + d)$, and this is the group of orientation-preserving isometries of the hyperbolic metric.\nA subgroup $\\Gamma \\subseteq \\mathrm{PSL}_2(\\R)$ is Fuchsian if and only if its action on $\\mathbb H$ is [[properly discontinuous|J5X5ED0B2Y]].\n\nExample: $\\mathrm{PSL}_2(\\Z)$, the image of the integer matrices of determinant $1$, is Fuchsian.\nTwo distinct integer matrices differ by at least $1$ in some entry, so a small neighbourhood of $\\pm I$ contains no other element of the group.\n\nNon-example: the group generated by $z \\mapsto z + 1$ and $z \\mapsto z + \\sqrt2$ consists of the translations $z \\mapsto z + a + b\\sqrt2$ with $a, b \\in \\Z$.\nThe translation lengths $a + b\\sqrt2$ come arbitrarily close to $0$ without being $0$, so the group is not discrete, and it is not Fuchsian.\n\nSome authors call a group Fuchsian when it is a discrete group of Möbius transformations that preserves a disc or half-plane in the Riemann sphere.\nSuch a group is conjugate to a group as above.\nSince $\\mathrm{PSL}_2(\\R) \\subset \\mathrm{PSL}_2(\\C)$, every Fuchsian group is a Kleinian group.\n",
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