Algebraic & Geometric Topology

Hyperplanes of Squier's cube complexes

Anthony Genevois

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


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.

Article information

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

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

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx] 20F67: Hyperbolic groups and nonpositively curved groups

diagram groups CAT(0) cube complexes special groups Squier complexes right-angled Artin groups


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

Export citation


  • I Agol, The virtual Haken conjecture, Doc. Math. 18 (2013) 1045–1087
  • G N Arzhantseva, V S Guba, M V Sapir, Metrics on diagram groups and uniform embeddings in a Hilbert space, Comment. Math. Helv. 81 (2006) 911–929
  • W Ballmann, J Świ\katkowski, On groups acting on nonpositively curved cubical complexes, Enseign. Math. 45 (1999) 51–81
  • A Baudisch, Subgroups of semifree groups, Acta Math. Acad. Sci. Hungar. 38 (1981) 19–28
  • G Baumslag, J E Roseblade, Subgroups of direct products of free groups, J. London Math. Soc. 30 (1984) 44–52
  • M R Bridson, A Haefliger, Metric spaces of non-positive curvature, Grundl. Math. Wissen. 319, Springer (1999)
  • S Buyalo, A Dranishnikov, V Schroeder, Embedding of hyperbolic groups into products of binary trees, Invent. Math. 169 (2007) 153–192
  • M Casals-Ruiz, A Duncan, I Kazachkov, Embedddings between partially commutative groups: two counterexamples, J. Algebra 390 (2013) 87–99
  • R Charney, M W Davis, The $K(\pi,1)$–problem for hyperplane complements associated to infinite reflection groups, J. Amer. Math. Soc. 8 (1995) 597–627
  • J Crisp, M Sageev, M Sapir, Surface subgroups of right-angled Artin groups, Internat. J. Algebra Comput. 18 (2008) 443–491
  • J Crisp, B Wiest, Embeddings of graph braid and surface groups in right-angled Artin groups and braid groups, Algebr. Geom. Topol. 4 (2004) 439–472
  • D S Farley, Finiteness and $\rm CAT(0)$ properties of diagram groups, Topology 42 (2003) 1065–1082
  • A Genevois, Hyperbolic diagram groups are free, Geom. Dedicata 188 (2017) 33–50
  • V Guba, M Sapir, Diagram groups, Mem. Amer. Math. Soc. 620, Amer. Math. Soc., Providence, RI (1997)
  • V S Guba, M V Sapir, On subgroups of the R Thompson group $F$ and other diagram groups, Mat. Sb. 190 (1999) 3–60 In Russian; translated in Sb. Math. 190 (1999) 1077–1130
  • V S Guba, M V Sapir, Diagram groups and directed $2$–complexes: homotopy and homology, J. Pure Appl. Algebra 205 (2006) 1–47
  • V S Guba, M V Sapir, Diagram groups are totally orderable, J. Pure Appl. Algebra 205 (2006) 48–73
  • M F Hagen, D T Wise, Cubulating hyperbolic free-by-cyclic groups: the general case, Geom. Funct. Anal. 25 (2015) 134–179
  • F Haglund, Finite index subgroups of graph products, Geom. Dedicata 135 (2008) 167–209
  • F Haglund, D T Wise, Special cube complexes, Geom. Funct. Anal. 17 (2008) 1551–1620
  • D Hume, Embedding mapping class groups into finite products of trees, preprint (2012)
  • V Kilibarda, On the algebra of semigroup diagrams, Internat. J. Algebra Comput. 7 (1997) 313–338
  • J Lauer, D T Wise, Cubulating one-relator groups with torsion, Math. Proc. Cambridge Philos. Soc. 155 (2013) 411–429
  • J M Mackay, A Sisto, Embedding relatively hyperbolic groups in products of trees, Algebr. Geom. Topol. 13 (2013) 2261–2282
  • T A McKee, F R McMorris, Topics in intersection graph theory, Society for Industrial and Applied Mathematics, Philadelphia (1999)
  • G A Niblo, L D Reeves, Coxeter groups act on $\mathrm{CAT}(0)$ cube complexes, J. Group Theory 6 (2003) 399–413
  • G A Niblo, M A Roller, Groups acting on cubes and Kazhdan's property $({\mathrm{T}})$, Proc. Amer. Math. Soc. 126 (1998) 693–699
  • A Y Ol'shanskii, M V Sapir, Length and area functions on groups and quasi-isometric Higman embeddings, Internat. J. Algebra Comput. 11 (2001) 137–170
  • S D Pauls, The large scale geometry of nilpotent Lie groups, Comm. Anal. Geom. 9 (2001) 951–982
  • M Sageev, Ends of group pairs and non-positively curved cube complexes, Proc. London Math. Soc. 71 (1995) 585–617
  • M Sageev, $\mathrm{CAT}(0)$ cube complexes and groups, from “Geometric group theory” (M Bestvina, M Sageev, K Vogtmann, editors), IAS/Park City Math. Ser. 21, Amer. Math. Soc., Providence, RI (2014) 7–54
  • P Scott, T Wall, Topological methods in group theory, from “Homological group theory” (C T C Wall, editor), London Math. Soc. Lecture Note Ser. 36, Cambridge Univ. Press (1979) 137–203
  • J-P Serre, Trees, Springer (2003)
  • D T Wise, Cubulating small cancellation groups, Geom. Funct. Anal. 14 (2004) 150–214