Algebraic & Geometric Topology

Hyperplanes of Squier's cube complexes

Anthony Genevois

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Abstract

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

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

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

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

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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References

  • 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