Quadric Complexes

Abstract

Quadric complexes are square complexes satisfying a certain combinatorial nonpositive curvature condition. These complexes generalize $2$-dimensional $CAT\left(0\right)$ 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 $\mathrm{C}\left(4\right)\text{-}\mathrm{T}\left(4\right)$ small cancelation groups act on quadric complexes.