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

We study quotients $\Gamma\backslash \mathbb H^n$ of the $n$-fold product of the upper half plane $\mathbb H$ by irreducible and torsion-free lattices $\Gamma < PSL_2(\mathbb R)^n$ 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 $(\mathbb P^1)^n$ of the form $\Gamma\backslash \mathbb H^n$ with $\Gamma$ contained in the norm-1 group of a maximal order of a quaternion algebra over a real number field.