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  "body": "\n## Example\n\nLet $L = \\bigoplus_{i \\in \\Z} \\Z/2\\Z$ be the group of finitely supported functions $f \\colon \\Z \\to \\Z/2\\Z$, with pointwise addition.\nLet $\\Z$ act on $L$ by shifts: $(n \\cdot f)(i) = f(i - n)$.\nThe *lamplighter group* is the [[semidirect product|0H3WZXCZNF]]\n$$\\Z/2\\Z \\wr \\Z = L \\rtimes \\Z,$$\nthe restricted [[wreath product|W1X984PKJ9]] of $\\Z/2\\Z$ by $\\Z$.\nAn element is a pair $(f, n)$.\nHere $f$ says which lamps on the integer line are lit, finitely many of them, and $n$ is the position of the lamplighter.\n\nIt is generated by $t = (0, 1)$, which moves the lamplighter one step, and $a = (\\delta_0, 0)$, which switches the lamp at $0$.\nSo it is a [[finitely generated group|082DWWXMRJ]].\nIt is an extension of the abelian group $L$ by the abelian group $\\Z$, so it is [[solvable|0FXJ7YFMMX]] of derived length $2$.\nSolvable groups are amenable, so it is [[amenable|CEAT9BESKK]].\n\n## Remarks\n\nThe group has exponential [[growth|4CMRM3975W]].\nConsider the $2^n$ words $a^{\\varepsilon_1} t \\, a^{\\varepsilon_2} t \\cdots a^{\\varepsilon_n} t$ with $\\varepsilon_i \\in \\set{0, 1}$, of length at most $2n$.\nThey give distinct elements, because the lamps lit along the way, at $n$ consecutive positions, record the sequence $(\\varepsilon_1, \\ldots, \\varepsilon_n)$.\nSo it is a solvable group that is not virtually nilpotent.\nIt has the infinite presentation $\\langle a, t \\mid a^2, \\ [a, t^k a t^{-k}] \\text{ for } k \\in \\Z \\rangle$, and it is not finitely presented.\n\n## Explanations\n\n### L1 human\n\nPicture an infinite street of lamps, one at each integer, and a lamplighter walking along it.\nA state is a finite set of lit lamps and a position of the lamplighter.\nThe two moves are: walk one step, or switch the lamp where you stand.\nCombining moves is the group operation.\nThis simple group is a standard source of examples and counterexamples in [[geometric group theory|2T3218RXTD]].\n",
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