I have a (maybe too naive) hope that cocompact torsion-free arithmetic lattices in hyperbolic spaces $X \neq \mathbb{H}_\mathbb{R}^2$ are uniquely determined by their cohomology with coefficients in $\mathbb{Z}$. I was wondering if there are any known counter examples to this claim, that is:

**Question**: Are there torsion free cocompact lattices of hyperbolic spaces (real, complex or quaternionic of dimension $\geq 3$) that are not isomorphic but have the same cohomolgy?