Hyperplanes of Squier's cube complexes

Abstract

To any semigroup presentation $\mathcal{P}=\langle \mathrm{\Sigma }\mid \mathcal{ℛ}\rangle$ and base word $w\in {\mathrm{\Sigma }}^{+}$ may be associated a nonpositively curved cube complex $S\left(\mathcal{P},w\right)$, called a Squier complex, whose underlying graph consists of the words of ${\mathrm{\Sigma }}^{+}$ equal to $w$ modulo $\mathcal{P}$, where two such words are linked by an edge when one can be transformed into the other by applying a relation of $\mathcal{ℛ}$. 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 $S\left(\mathcal{P},w\right)$ 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 $\prec$. As a strong consequence on the geometry of $S\left(\mathcal{P},w\right)$, 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 $S\left(\mathcal{P},w\right)$ 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 $\mathcal{P}\left(\mathrm{\Gamma }\right)$ to any finite interval graph $\mathrm{\Gamma }$, and we prove that the diagram group associated to $\mathcal{P}\left(\mathrm{\Gamma }\right)$ (for a given base word) is isomorphic to the right-angled Artin group $A\left(\bar{\mathrm{\Gamma }}\right)$. This result has many consequences on the study of subgroups of diagram groups. In particular, we deduce that, for all $n\ge 1$, the right-angled Artin group $A\left(C_n\right)$ embeds into a diagram group, answering a question of Guba and Sapir.