Varieties of general type with the same Betti numbers as ${\mathbb{P}}^1\times {\mathbb{P}}^1\times \dots \times {\mathbb{P}}^1$

We study quotients $\mathrm{\Gamma }\mathrm{\backslash }{\mathbb{H}}^n$ of the $n$-fold product of the upper half plane $\mathbb{H}$ by irreducible and torsion-free lattices $\mathrm{\Gamma }<PS{L}_2(\mathbb{R}{)}^n$ with the same Betti numbers as the $n$-fold product $({\mathbb{P}}^1{)}^n$ of projective lines. Such varieties are called fake products of projective lines or fake $({\mathbb{P}}^1{)}^n$. These are higher dimensional analogs of fake quadrics. In this paper we show that the number of fake $({\mathbb{P}}^1{)}^n$ is finite (independently of $n$), we give examples of fake $({\mathbb{P}}^1{)}^4$ and show that for $n>4$ there are no fake $(\mathbb P^1)^n$oftheform$\Gamma\backslash \mathbb H^n$with$\Gamma$ contained in the norm-1 group of a maximal order of a quaternion algebra over a real number field.