  In the present article, we provide examples of fake quadrics, that is,
minimal complex surfaces of general type with the same numerical invariants as
the smooth quadric in $\PP ^3$ which are quotients of the bidisc by an
irreducible lattice of automorphisms. Moreover, we list classes of arithmetic
lattices over a real quadratic number field which define a fake quadric and
give general results towards a classification of all such fake quadrics.

Algebraic Geometry;Geometric Topology