Non-positively Curved Cube Complexes

Let Γ be a discrete group, defined by a presentation P = 〈ai | rj〉, say, or as the fundamental group of a connected CW-complex X. Remark 1.1. Let XP be the CW-complex with a single 0-cell E , one 1-cell E i for each ai (oriented accordingly), and one 2-cell E 2 j for each rj , with attaching map ∂E j → X (1) P that reads off the word rj in the generators {ai}. Then, by the Seifert–van Kampen Theorem, the fundamental group of XP is the group presented by P. Conversely, restricting attention to X and contracting a maximal tree in X, we obtain XP for some presentation P of Γ. So we see that these two situations are equivalent. We will call XP a presentation complex for Γ. Here are two typical questions that we might want to be able to answer about Γ.