Hyperplanes of squier’s cube complexes

Abstract

To any semigroup presentation P=〈Σ|R〉 and base word w ∈ Σ+ may be associated a nonpositively curved cube complex S(P, w), called a Squier complex, whose underlying graph consists of the words of Σ+ equal to w modulo P, where two such words are linked by an edge when one can be transformed into the other by applying a relation of R. A group is a diagram group if it is the fundamental group of a Squier complex. We describe hyperplanes in these cube complexes. As a first application, we determine exactly when S(P,w) is a special cube complex, as defined by Haglund and Wise, so that the associated diagram group embeds into a right-angled Artin group. A particular feature of Squier complexes is that the intersections of hyperplanes are “ordered” by a relation (formula presented). As a strong consequence on the geometry of S(P,w), we deduce, in finite dimensions, that its universal cover isometrically embeds into a product of finitely many trees with respect to the combinatorial metrics; in particular, we notice that (often) this allows us to embed quasi-isometrically the associated diagram group into a product of finitely many trees, giving information on its asymptotic dimension and its uniform Hilbert space compression. Finally, we exhibit a class of hyperplanes inducing a decomposition of S(P,w) as a graph of spaces, and a fortiori a decomposition of the associated diagram group as a graph of groups, giving a new method to compute presentations of diagram groups. As an application, we associate a semigroup presentation P(Γ) to any finite interval graph Γ and we prove that the diagram group associated to P(Γ) (for a given base word) is isomorphic to the right-angled Artin group (formula presented) This result has many consequences on the study of subgroups of diagram groups. In particular, we deduce that, for all n ≥ 1, the right-angled Artin group A(Cn) embeds into a diagram group, answering a question of Guba and Sapir.