  Quadric complexes are square complexes satisfying a certain combinatorial
nonpositive curvature condition. These complexes generalize 2-dimensional
CAT(0) cube complexes and are a square analog of systolic complexes. We
introduce and study the basic properties of these complexes. Using a form of
dismantlability for the 1-skeleta of finite quadric complexes we show that
every finite group acting on a quadric complex stabilizes a complete bipartite
subgraph of its 1-skeleton. Finally, we prove that C(4)-T(4) small cancellation
groups act on quadric complexes.
