Inoue surface

Example

Let $M$ be a matrix in the special linear group $\mathrm{SL}(3, \Z)$ with one real eigenvalue $\alpha > 1$ and two non-real eigenvalues $\beta, \bar\beta$. Let $a = (a_1, a_2, a_3)^T \in \R^3$ be an eigenvector of $M$ for $\alpha$ and $b = (b_1, b_2, b_3)^T \in \C^3$ an eigenvector for $\beta$. Let $\mathbb{H}$ be the upper half-plane and let $G_M$ be the group of holomorphic automorphisms of $\mathbb{H} \times \C$ generated by $$g_0(w, z) = (\alpha w, \beta z), \qquad g_i(w, z) = (w + a_i, z + b_i) \quad (i = 1, 2, 3).$$ The quotient $S_M = (\mathbb{H} \times \C)/G_M$ is the Inoue surface of type $S_M$, a compact complex surface.

Check 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 $\beta b_i = \sum_j m_{ij} b_j$. Hence $g_0 g_i g_0^{-1} = g_1^{m_{i1}} g_2^{m_{i2}} g_3^{m_{i3}}$, so the translations form a normal subgroup isomorphic to $\Z^3$, and $G_M$ is a semidirect product $\Z^3 \rtimes \Z$. The maps $g_i$ keep $\operatorname{Im} w$ fixed, because $a_i$ is real, and $g_0$ multiplies it by $\alpha$; so $G_M$ preserves $\mathbb{H} \times \C$.

A concrete $M$: the companion matrix $M = \begin{pmatrix} 0 & 0 & 1 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix}$ of $x^3 - x - 1$ has determinant $1$. The discriminant of $x^3 - x - 1$ is $-4(-1)^3 - 27 = -23 < 0$, so it has one real root $\alpha \approx 1.3247 > 1$ and two non-real roots.