#### Volume 18, issue 6 (2018)

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Hyperplanes of Squier's cube complexes

### Anthony Genevois

Algebraic & Geometric Topology 18 (2018) 3205–3256
##### Abstract

To any semigroup presentation $\mathsc{P}=〈\Sigma \mid \mathsc{ℛ}〉$ and base word $w\in {\Sigma }^{+}$ may be associated a nonpositively curved cube complex $S\left(\mathsc{P},w\right)$, called a Squier complex, whose underlying graph consists of the words of ${\Sigma }^{+}$ equal to $w$ modulo $\mathsc{P}$, where two such words are linked by an edge when one can be transformed into the other by applying a relation of $\mathsc{ℛ}$. 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 $S\left(\mathsc{P},w\right)$ 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 $S\left(\mathsc{P},w\right)$, 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 $S\left(\mathsc{P},w\right)$ 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 $\mathsc{P}\left(\Gamma \right)$ to any finite interval graph $\Gamma$, and we prove that the diagram group associated to $\mathsc{P}\left(\Gamma \right)$ (for a given base word) is isomorphic to the right-angled Artin group $A\left(\stackrel{̄}{\Gamma }\right)$. 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.

##### Keywords
diagram groups, CAT(0) cube complexes, special groups, Squier complexes, right-angled Artin groups
##### Mathematical Subject Classification 2010
Primary: 20F65, 20F67