The intersection form on the group of line bundles on a complex algebraic surface always has signature $(1,n)$ for some $n$. So the automorphism group of an algebraic surface always acts on hyperbolic $n$-space. For a class of surfaces including $K3$ surfaces and many rational surfaces, there is a close connection between the properties of the variety and the corresponding group acting on hyperbolic space. (In fancier terms: the Morrison-Kawamata cone conjecture holds for klt Calabi-Yau pairs in dimension 2.)