Inventiones mathematicae

, Volume 213, Issue 3, pp 1179–1247 | Cite as

Commensurability of groups quasi-isometric to RAAGs

  • Jingyin HuangEmail author


Let G be a right-angled Artin group with defining graph \(\Gamma \) and let H be a finitely generated group quasi-isometric to G. We show if G satisfies that (1) its outer automorphism group is finite; (2) \(\Gamma \) does not contain any induced 4-cycles; (3) \(\Gamma \) is star-rigid; then H is commensurable to G. We show condition (2) is sharp in the sense that if \(\Gamma \) contains an induced 4-cycle, then there exists an H quasi-isometric to G but not commensurable to G. Moreover, one can drop condition (1) if H is a uniform lattice acting on the universal cover of the Salvetti complex of G. As a consequence, we obtain a conjugation theorem for such uniform lattices. The ingredients of the proof include a blow-up building construction in Huang and Kleiner (Duke Math. J. 167(3), 537-602 (2018). and a Haglund–Wise style combination theorem for certain class of special cube complexes. However, in most of our cases, relative hyperbolicity is absent, so we need new ingredients for the combination theorem.

Mathematics Subject Classification

20F65 20F67 20F69 



This paper would not be possible without the helpful discussions with B. Kleiner. In particular, he pointed out a serious gap in the author’s previous attempt to prove a special case of the main theorem. Also the author learned Lemma 4.11 from him. The author thanks D. T. Wise for pointing out the reference [35] and X. Xie for helpful comments and clarifications. The author thanks the referee for carefully reading the paper and providing many helpful comments and clarifications.


  1. 1.
    Abramenko, P., Brown, K.S.: Buildings: Theory and Applications. Springer, Berlin (2008)CrossRefzbMATHGoogle Scholar
  2. 2.
    Agol, I., Groves, D., Manning, J.: The virtual haken conjecture. Doc. Math. 18, 1045–1087 (2013)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bass, H.: The degree of polynomial growth of finitely generated nilpotent groups. Proc. Lond. Math. Soc. 3(4), 603–614 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bass, H., Kulkarni, R.: Uniform tree lattices. J. Am. Math. Soc. 3, 843–902 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Behrstock, J., Januszkiewicz, T., Neumann, W.: Commensurability and QI classification of free products of finitely generated abelian groups. Proc. Am. Math. Soc. 137(3), 811–813 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Behrstock, J., Kleiner, B., Minsky, Y., Mosher, L.: Geometry and rigidity of mapping class groups. Geom. Topol. 16(2), 781–888 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Behrstock, J.A., Januszkiewicz, T., Neumann, W.D.: Quasi-isometric classification of some high dimensional right-angled Artin groups. Groups Geom. Dyn. 4(4), 681–692 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Behrstock, J.A., Neumann, W.D.: Quasi-isometric classification of graph manifold groups. Duke Math. J. 141(2), 217–240 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bestvina, M., Kleiner, B., Sageev, M.: The asymptotic geometry of right-angled Artin groups I. Geom. Topol. 12(3), 1653–1699 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Bestvina, M., Kleiner, B., Sageev, M.: Quasiflats in CAT(0) 2-complexes. Algebr. Geom. Topol. 16(5), 2663–2676 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bou-Rabee, K., Hagen, M.F., Patel, P.: Residual finiteness growths of virtually special groups. Mathematische Zeitschrift 279(1–2), 297–310 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Bowditch, B.H.: Large-scale rank and rigidity of the Teichmüller metric. J. Topol. 9(4), 985–1020 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Bowditch, B.: Large-scale rank and rigidity of the Weil–Petersson metric (2015). PreprintGoogle Scholar
  14. 14.
    Bowditch, B.H.: Large-scale rigidity properties of the mapping class groups. Pacific J. Math. 293(1), 1–73 (2018). MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Bridson, M.R., Haefliger, A.: Metric Spaces of Non-positive Curvature, vol. 319. Springer, Berlin (1999)zbMATHGoogle Scholar
  16. 16.
    Bridson, M.R., Haefliger, A.: Metric Spaces of Non-positive Curvature, Volume 319 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319. Springer, Berlin (1999)Google Scholar
  17. 17.
    Burger, M., Mozes, S.: Lattices in product of trees. Inst. Hautes Études Sci. Publ. Math. 92(2000), 151–194 (2001)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Burger, M., Mozes, S.: Lattices in product of trees. Publications Mathématiques de l’IHÉS 92, 151–194 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Caprace, P.-E., Sageev, M.: Rank rigidity for CAT (0) cube complexes. Geom. Funct. Anal. 21(4), 851–891 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Charney, R.: An introduction to right-angled Artin groups. Geom. Dedic. 125(1), 141–158 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Charney, R., Farber, M.: Random groups arising as graph products. Algebraic Geom. Topol. 12(2), 979–995 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Davis, M.W.: Buildings are cat (0). Geom. Cohomol. Group Theory (Durham 1994) 252, 108–123 (1994)MathSciNetGoogle Scholar
  23. 23.
    Day, M.B.: Finiteness of outer automorphism groups of random right-angled artin groups. Algebraic Geom. Topol. 12(3), 1553–1583 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Droms, C.: Isomorphisms of graph groups. Proc. Am. Math. Soc. 100(3), 407–408 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Dunwoody, M.J.: The accessibility of finitely presented groups. Invent. Math. 81(3), 449–457 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Eskin, A.: Quasi-isometric rigidity of nonuniform lattices in higher rank symmetric spaces. J. Am. Math. Soc. 11(2), 321–361 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Eskin, A., Farb, B.: Quasi-flats and rigidity in higher rank symmetric spaces. J. Am. Math. Soc. 10(3), 653–692 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Eskin, A., Masur, H., Rafi, K.: Rigidity of teichmüller space (2015). Preprint arXiv:1506.04774
  29. 29.
    Frigerio, R., Lafont, J.-F., Sisto, A.: Rigidity of high dimensional graph manifolds. Astérisque 372 (2015). ISBN 978-2-85629-809-1Google Scholar
  30. 30.
    Gromov, M.: Groups of polynomial growth and expanding maps (with an appendix by jacques tits). Publications Mathématiques de l’IHÉS 53, 53–78 (1981)CrossRefzbMATHGoogle Scholar
  31. 31.
    Gromov, M.: Asymptotic invariants of infinite groups. In: Niblo, A., Roller, M.A. (eds.) Geometric Group Theory, vol. 2. (Sussex, 1991). London Mathematical Society Lecture Note Series, vol. 182, pp. 1–295. Cambridge University Press, Cambridge (1993)Google Scholar
  32. 32.
    Hagen, M.F., Przytycki, P.: Cocompactly cubulated graph manifolds. Isr. J. Math. 207, 1–18 (2013)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Hagen, M.F., Wise, D.T.: Cubulating hyperbolic free-by-cyclic groups: the irreducible case. Duke Math. J. 165(9), 1753–1813 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Hagen, M.F., Wise, D.T.: Cubulating hyperbolic free-by-cyclic groups: the general case. Geom. Funct. Anal. 25(1), 134–179 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Haglund, F.: Commensurability and separability of quasiconvex subgroups. Algebraic Geom. Topol. 6(2), 949–1024 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Haglund, F.: Finite index subgroups of graph products. Geom. Dedic. 135(1), 167–209 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Haglund, F., Wise, D.T.: Special cube complexes. Geom. Funct. Anal. 17(5), 1551–1620 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Haglund, F., Wise, D.T.: Coxeter groups are virtually special. Adv. Math. 224(5), 1890–1903 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Haglund, F., Wise, D.T.: A combination theorem for special cube complexes. Ann. Math. 176(3), 1427–1482 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Hamenstaedt, U.: Geometry of the mapping class groups III: quasi-isometric rigidity (2005). Preprint arXiv:math/0512429
  41. 41.
    Huang, J.: Quasi-isometric classification of right-angled Artin groups I: the finite out case. Geom. Topol. 21(6), 3467–3537 (2017). MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Huang, J.: Top-dimensional quasiflats in CAT(0) cube complexes. Geom. Topol. 21(4), 2281–2352 (2017). MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Huang, J., Jankiewicz, K., Przytycki, P.: Cocompactly cubulated 2-dimensional Artin groups. Comment. Math. Helv. 91(3), 519–542 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Huang, J., Kleiner, B.: Groups quasi-isometric to right-angled Artin groups. Duke Math. J. 167(3), 537–602 (2018). MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Janzen, D., Wise, D.T.: A smallest irreducible lattice in the product of trees. Algebraic Geom. Topol. 9(4), 2191–2201 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Kapovich, M., Leeb, B.: Quasi-isometries preserve the geometric decomposition of Haken manifolds. Invent. Math. 128(2), 393–416 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Karrass, A., Pietrowski, A., Solitar, D.: Finite and infinite cyclic extensions of free groups. J. Aust. Math. Soc. 16(04), 458–466 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Kim, S.-H., Koberda, T.: Embedability between right-angled Artin groups. Geom. Topol. 17(1), 493–530 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Kleiner, B.: The local structure of length spaces with curvature bounded above. Mathematische Zeitschrift 231(3), 409–456 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Kleiner, B., Leeb, B.: Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings. Comptes Rendus de l’Académie des Sciences-Series I-Mathematics 324(6), 639–643 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Liu, Y.: Virtual cubulation of nonpositively curved graph manifolds. J. Topol. 6, 793–822 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Ollivier, Y., Wise, D.: Cubulating random groups at density less than 1/6. Trans. Am. Math. Soc. 363(9), 4701–4733 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Przytycki, P., Wise, D.T.: Mixed 3-manifolds are virtually special (2012). Preprint arXiv:1205.6742
  54. 54.
    Przytycki, P., Wise, D.T.: Graph manifolds with boundary are virtually special. J. Topol. 7, 419–435 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Scott, P.: Subgroups of surface groups are almost geometric. J. Lond. Math. Soc. 2(3), 555–565 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    Stallings, J.R.: On torsion-free groups with infinitely many ends. Ann. Math. 88, 312–334 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Stallings, J.R.: Topology of finite graphs. Invent. Math. 71(3), 551–565 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    Whyte, K.: Coarse bundles (2010). Preprint arXiv:1006.3347
  59. 59.
    Wise, D.: The structure of groups with a quasiconvex hierarchy. Preprint (2011)Google Scholar
  60. 60.
    Wise, D.T.: Non-positively Curved Squared Complexes Aperiodic Tilings and Non-residually Finite Groups. Princeton University, Princeton (1996)Google Scholar
  61. 61.
    Wise, D.T.: The residual finiteness of negatively curved polygons of finite groups. Invent. Math. 149(3), 579–617 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  62. 62.
    Wise, D.T.: Cubulating small cancellation groups. Geom. Funct. Anal. GAFA 14(1), 150–214 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  63. 63.
    Wise, D.T.: Research announcement: the structure of groups with a quasiconvex hierarchy. Electron. Res. Announc. Math. Sci 16, 44–55 (2009)MathSciNetzbMATHGoogle Scholar
  64. 64.
    Wise, D.T.: From Riches to Raags: 3-Manifolds, Right-angled Artin Groups, and Cubical Geometry, vol. 117. American Mathematical Society, Providence (2012)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.The Department of Mathematics and StatisticsMcGill UniversityMontrealCanada

Personalised recommendations