Hyperplanes of Squier's cube complexes

TL;DR

This paper studies hyperplanes in Squier's cube complexes, characterizes when they are special, and explores their geometric and algebraic implications, including embeddings into products of trees and relations to right-angled Artin groups.

Contribution

It provides a detailed description of hyperplanes in Squier complexes, characterizes when these complexes are special, and links diagram groups to right-angled Artin groups via hyperplane analysis.

Findings

Characterization of hyperplanes in Squier complexes.

Criteria for Squier complexes to be special cube complexes.

Embedding of universal covers into products of trees and diagram groups into products of trees.

Abstract

To any semigroup presentation $\mathcal{P}= \langle \Sigma \mid \mathcal{R} \rangle$ and base word $w \in \Sigma^+$ may be associated a nonpositively curved cube complex $S(\mathcal{P},w)$, called a Squier complex, whose underlying graph consists of the words of $\Sigma^+$ 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. In this paper, we describe hyperplanes in these cube complexes. As a first application, we determine exactly when $S(\mathcal{P},w)$ 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 $\prec$. As a strong consequence on the geometry of $S(\mathcal{P},w)$, we deduce, in finite dimensions, that its univeral cover isometrically embedds into a product of finitely-many trees with respect to the combinatorial metrics; in particular, we notice that (often) this allows to embed quasi-isometrically the associated diagram group into a product of finitely-many trees. Finally, we exhibit a class of hyperplanes inducing a decomposition of $S(\mathcal{P},w)$ 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 $\mathcal{P}(\Gamma)$ to any finite interval graph $\Gamma$, and we prove that the diagram group associated to $\mathcal{P}(\Gamma)$ (for a given base word) is isomorphic to the right-angled Artin group $A(\overline{\Gamma})$.