Quadric Complexes

TL;DR

Quadric complexes are a class of square complexes with nonpositive curvature properties, generalizing CAT(0) cube complexes and systolic complexes, with applications to group actions and small cancellation groups.

Contribution

This paper introduces the concept of quadric complexes, explores their properties, and demonstrates their relevance to group actions and small cancellation groups.

Findings

Finite groups acting on quadric complexes stabilize bipartite subgraphs.

C(4)-T(4) small cancellation groups act on quadric complexes.

Quadric complexes generalize known nonpositive curvature complexes.

Abstract

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.