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  "body": "\n## Definition\n\nLet $g \\ge 1$ be an integer.\nLet $D_1, D_1', \\ldots, D_g, D_g'$ be $2g$ open Jordan discs in the [[Riemann sphere|0J78KBPFZB]] $\\hat{\\C}$ whose closures are pairwise disjoint.\nLet $C_i$ and $C_i'$ be the boundary curves of $D_i$ and $D_i'$.\nFor each $i$, let $A_i$ be a [[Möbius transformation|0SG74WBFA2]] that maps $C_i$ onto $C_i'$ and maps $\\hat{\\C} \\setminus \\overline{D_i}$ onto $D_i'$.\nThe group generated by $A_1, \\ldots, A_g$ is a *Schottky group* of genus $g$.\nIt is a *classical* Schottky group if the curves $C_i$ and $C_i'$ can be chosen to be circles.\n\n## Remarks\n\nExample: let $\\lambda \\in \\C$ with $\\abs{\\lambda} \u003e 1$, $A_1(z) = \\lambda z$, $D_1 = \\set{\\abs{z} \u003c 1}$ and $D_1' = \\set{\\abs{z} \u003e \\abs{\\lambda}} \\cup \\set{\\infty}$.\nTheir closures are disjoint, and $A_1$ maps the circle $\\abs{z} = 1$ onto $\\abs{z} = \\abs{\\lambda}$ and the region $\\abs{z} \u003e 1$ onto $D_1'$.\nSo $\\langle A_1 \\rangle$ is a Schottky group of genus $1$; the quotient of $\\C \\setminus \\set{0}$ by it is a torus.\n\nNon-example: the group generated by $z \\mapsto z + 1$ is not a Schottky group.\nIts generator is parabolic, while every element other than the identity of a Schottky group is loxodromic.\n\nA Schottky group of genus $g$ is a [[free group|0KTW26C80J]] of rank $g$ on $A_1, \\ldots, A_g$ and is a Kleinian group.\nThe common exterior of the $2g$ discs is a fundamental domain for its action on the domain of discontinuity $\\Omega$.\nThe quotient $\\Omega / G$ is a compact Riemann surface of genus $g$.\n",
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