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  "body": "\n## Example\n\nLet $M$ be a matrix in the [[special linear group|0V9PMN0RBH]] $\\mathrm{SL}(3, \\Z)$ with one real [[eigenvalue|0BQAA2BK3Q]] $\\alpha \u003e 1$ and two non-real eigenvalues $\\beta, \\bar\\beta$.\nLet $a = (a_1, a_2, a_3)^T \\in \\R^3$ be an eigenvector of $M$ for $\\alpha$ and $b = (b_1, b_2, b_3)^T \\in \\C^3$ an eigenvector for $\\beta$.\nLet $\\mathbb{H}$ be the [[upper half-plane|X6XGGH5FPW]] and let $G_M$ be the group of holomorphic automorphisms of $\\mathbb{H} \\times \\C$ generated by\n$$g_0(w, z) = (\\alpha w, \\beta z), \\qquad g_i(w, z) = (w + a_i, z + b_i) \\quad (i = 1, 2, 3).$$\nThe quotient $S_M = (\\mathbb{H} \\times \\C)/G_M$ is the *Inoue surface* of type $S_M$, a compact complex surface.\n\nCheck of the group structure: from $M a = \\alpha a$ and $M b = \\beta b$ one gets $\\alpha a_i = \\sum_j m_{ij} a_j$ and $\\beta b_i = \\sum_j m_{ij} b_j$.\nHence $g_0 g_i g_0^{-1} = g_1^{m_{i1}} g_2^{m_{i2}} g_3^{m_{i3}}$, so the translations form a normal subgroup isomorphic to $\\Z^3$, and $G_M$ is a semidirect product $\\Z^3 \\rtimes \\Z$.\nThe maps $g_i$ keep $\\operatorname{Im} w$ fixed, because $a_i$ is real, and $g_0$ multiplies it by $\\alpha$; so $G_M$ preserves $\\mathbb{H} \\times \\C$.\n\nA concrete $M$: the companion matrix $M = \\begin{pmatrix} 0 \u0026 0 \u0026 1 \\\\ 1 \u0026 0 \u0026 1 \\\\ 0 \u0026 1 \u0026 0 \\end{pmatrix}$ of $x^3 - x - 1$ has determinant $1$.\nThe discriminant of $x^3 - x - 1$ is $-4(-1)^3 - 27 = -23 \u003c 0$, so it has one real root $\\alpha \\approx 1.3247 \u003e 1$ and two non-real roots.\n\n## Remarks\n\nThe map $(w, z) \\mapsto \\log \\operatorname{Im} w$ descends to a map $S_M \\to \\R / (\\log\\alpha)\\Z$, which makes $S_M$ a bundle over a circle with fibre a real $3$-torus.\nIts Betti numbers are $b_1 = 1$ and $b_2 = 0$, it contains no compact complex curves, and its Kodaira dimension is $-\\infty$, so it is a [[class VII surface|Y4NSSJT4RC]].\nIts universal cover is $\\mathbb{H} \\times \\C$, not $\\C^2 \\setminus \\set{0}$, so it is not a [[Hopf surface|PNFNGBXRM3]].\nInoue also introduced two further families, $S^{+}$ and $S^{-}$, built in a similar way from integer matrices.\nA theorem of Bogomolov states that a minimal class VII surface with $b_2 = 0$ is a Hopf surface or an Inoue surface.\nLi, Yau and Zheng, and later Teleman, gave complete proofs.\n\n## Explanations\n\n### L1 human\n\nAn Inoue surface is built from the product of the upper half-plane and the complex line, divided by a group made from an integer $3 \\times 3$ matrix.\nThe group shifts points in three independent directions and stretches by the eigenvalues of the matrix.\nThe result is a compact complex surface that contains no complex curves at all.\nTogether with the Hopf surfaces, these surfaces are the class VII surfaces whose second Betti number is zero.\n",
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