To any semigroup presentation and base word may be associated a nonpositively curved cube complex , called a Squier complex, whose underlying graph consists of the words of equal to modulo , where two such words are linked by an edge when one can be transformed into the other by applying a relation of . 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 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 . As a strong consequence on the geometry of , 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 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 to any finite interval graph , and we prove that the diagram group associated to (for a given base word) is isomorphic to the right-angled Artin group . This result has many consequences on the study of subgroups of diagram groups. In particular, we deduce that, for all , the right-angled Artin group embeds into a diagram group, answering a question of Guba and Sapir.
"Hyperplanes of Squier's cube complexes." Algebr. Geom. Topol. 18 (6) 3205 - 3256, 2018. https://doi.org/10.2140/agt.2018.18.3205