## Algebraic & Geometric Topology

### Hyperplanes of Squier's cube complexes

Anthony Genevois

#### 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 $≺$. 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 $A(Γ̄)$. 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.

#### Article information

Source
Algebr. Geom. Topol., Volume 18, Number 6 (2018), 3205-3256.

Dates
Received: 17 April 2017
Revised: 19 January 2018
Accepted: 23 April 2018
First available in Project Euclid: 27 October 2018

Permanent link to this document
https://projecteuclid.org/euclid.agt/1540605641

Digital Object Identifier
doi:10.2140/agt.2018.18.3205

Mathematical Reviews number (MathSciNet)
MR3868219

Zentralblatt MATH identifier
06990062

#### Citation

Genevois, Anthony. Hyperplanes of Squier's cube complexes. Algebr. Geom. Topol. 18 (2018), no. 6, 3205--3256. doi:10.2140/agt.2018.18.3205. https://projecteuclid.org/euclid.agt/1540605641

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