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2018 Hyperplanes of Squier's cube complexes
Anthony Genevois
Algebr. Geom. Topol. 18(6): 3205-3256 (2018). DOI: 10.2140/agt.2018.18.3205


To any semigroup presentation P=Σ and base word wΣ+ 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 . 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 n1, the right-angled Artin group A(Cn) embeds into a diagram group, answering a question of Guba and Sapir.


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Anthony Genevois. "Hyperplanes of Squier's cube complexes." Algebr. Geom. Topol. 18 (6) 3205 - 3256, 2018.


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

zbMATH: 06990062
MathSciNet: MR3868219
Digital Object Identifier: 10.2140/agt.2018.18.3205

Primary: 20F65, 20F67

Rights: Copyright © 2018 Mathematical Sciences Publishers


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Vol.18 • No. 6 • 2018
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