Geometriae Dedicata

, Volume 124, Issue 1, pp 191–198 | Cite as

Embedding right-angled Artin groups into graph braid groups

  • Lucas Sabalka
Original Paper


We construct an embedding of any right-angled Artin group G(Δ) defined by a graph Δ into a graph braid group. The number of strands required for the braid group is equal to the chromatic number of Δ. This construction yields an example of a hyperbolic surface subgroup embedded in a two strand planar graph braid group.


Right-angled Artin group Graph braid group Hyperbolic surface subgroup 

Mathematics Subject Classifications (2000)

Primary: 20F36 Secondary: 20F65 53R80 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abrams, A.: Configuration spaces of braid groups of graphs. PhD thesis, UC Berkeley (2000)Google Scholar
  2. 2.
    Abrams A., Ghrist R. (2002) Finding topology in a factory: configuration spaces. Amer. Math. Monthly 109(2):140–150zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bestvina M., Brady N. (1997) Morse theory and finiteness properties of groups. Invent. Math. 129(3):445–470zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bridson, M.R., and Haefliger, A.: Metric spaces of non-positive curvature, volume 319 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin (1999)Google Scholar
  5. 5.
    Connolly, F., and Doig, M.: Braid groups and right-angled Artin groups. Preprint, 2004; arxiv:math.GT/0411368.Google Scholar
  6. 6.
    Crisp J., Wiest B. (2004) Embeddings of graph braid and surface groups in right-angled Artin groups and braid groups. Algebra Geom. Topol. 4:439–472zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Davis M.W., and Januszkiewicz T. (2000) Right-angled Artin groups are commensurable with right-angled Coxeter groups. J. Pure Appl. Algebra 153(3):229–235zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Droms C. (1987) Subgroups of graph groups. J. Algebra 110(2):519–522zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Farber M. (2003) Topological complexity of motion planning. Discrete Comput. Geom. 29(2):211–221zbMATHMathSciNetGoogle Scholar
  10. 10.
    Farber M. (2004) Instabilities of robot motion. Topol. Appl. 140(2–3):245–266zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Farley, D., and Sabalka, L.: On the cohomology rings of graph braid groups. Preprint, 2006; arxiv:math.GR/0602444Google Scholar
  12. 12.
    Farley, D., and Sabalka, L.: Morse theory and graph braid groups. Algebra Geom. Topol. 5, 1075–1109 (2005) Available online at: Scholar
  13. 13.
    Ghrist, R., and Peterson, V.: The geometry and topology of reconfiguration. Preprint, (2005)Google Scholar
  14. 14.
    Ghrist, R.: Configuration spaces and braid groups on graphs in robotics. In Knots, Braids, and Mapping Class Groups—Papers dedicated to Joan S. Birman (New York, 1998), vol. 24 of AMS/IP Stud. Adv. Math. pp. 29–40. Amer. Math. Soc. Providence, RI (2001)Google Scholar
  15. 15.
    Gordon C. McA., Long D.D., Reid A.W. (2004) Surface subgroups of Coxeter and Artin groups. J. Pure Appl. Algebra 189(1–3):135–148zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Lyndon, R.C., Schupp, P.E.: Combinatorial Group Theory Springer-Verlag, Berlin (1977) (Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89).Google Scholar
  17. 17.
    Servatius H., Droms C., Servatius B. (1989) Surface subgroups of graph groups. Proc. Amer. Math. Soc. 106(3):573–578zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Stallings J. (1963) A finitely presented group whose 3-dimensional integral homology is not finitely generated. Amer. J. Math. 85:541–543zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science + Business Media B.V. 2006

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CaliforniaDavisUSA

Personalised recommendations