Duke Mathematical Journal

Groups quasi-isometric to right-angled Artin groups

Jingyin Huang and Bruce Kleiner

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Abstract

We characterize groups quasi-isometric to a right-angled Artin group (RAAG) G with finite outer automorphism group. In particular, all such groups admit a geometric action on a CAT(0) cube complex that has an equivariant “fibering” over the Davis building of G. This characterization will be used in forthcoming work of the first author to give a commensurability classification of the groups quasi-isometric to certain RAAGs.

Article information

Source
Duke Math. J., Volume 167, Number 3 (2018), 537-602.

Dates
Received: 14 March 2016
Revised: 31 July 2017
First available in Project Euclid: 9 January 2018

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1515467194

Digital Object Identifier
doi:10.1215/00127094-2017-0042

Mathematical Reviews number (MathSciNet)
MR3761106

Zentralblatt MATH identifier
06848179

Subjects
Primary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx]
Secondary: 20F69: Asymptotic properties of groups

Keywords
cube complex right-angled Artin group quasi-isometry building rigidity

Citation

Huang, Jingyin; Kleiner, Bruce. Groups quasi-isometric to right-angled Artin groups. Duke Math. J. 167 (2018), no. 3, 537--602. doi:10.1215/00127094-2017-0042. https://projecteuclid.org/euclid.dmj/1515467194


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References

  • [1] P. Abramenko and K. S. Brown, Buildings: Theory and Applications, Grad. Texts in Math. 248, Springer, New York, 2008.
  • [2] I. Agol, The virtual Haken conjecture, Doc. Math. 18 (2013), 1045–1087.
  • [3] A. R. Ahlin, The large scale geometry of products of trees, Geom. Dedicata 92 (2002), 179–184.
  • [4] H. Bass, The degree of polynomial growth of finitely generated nilpotent groups, Proc. Lond. Math. Soc. (3) 25 (1972), 603–614.
  • [5] J. A. Behrstock, T. Januszkiewicz, and W. D. Neumann, Quasi-isometric classification of some high dimensional right-angled Artin groups, Groups Geom. Dyn. 4 (2010), 681–692.
  • [6] J. A. Behrstock and W. D. Neumann, Quasi-isometric classification of graph manifold groups, Duke Math. J. 141 (2008), 217–240.
  • [7] N. Bergeron and D. T. Wise, A boundary criterion for cubulation, Amer. J. Math. 134 (2012), 843–859.
  • [8] M. Bestvina and N. Brady, Morse theory and finiteness properties of groups, Invent. Math. 129 (1997), 445–470.
  • [9] M. Bestvina, B. Kleiner, and M. Sageev, The asymptotic geometry of right-angled Artin groups, I, Geom. Topol. 12 (2008), 1653–1699.
  • [10] M. Bestvina, B. Kleiner, and M. Sageev, Quasiflats in $\mathrm{CAT}(0)$ $2$-complexes, Algebr. Geom. Topol. 16 (2016), 2663–2676.
  • [11] N. Brady and J. Meier, Connectivity at infinity for right angled Artin groups, Trans. Amer. Math. Soc. 353, (2001), 117–132.
  • [12] M. R. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Grundlehren Math. Wissen. 319, Springer, Berlin, 1999.
  • [13] M. Burger and S. Mozes, Lattices in product of trees, Inst. Hautes Études Sci. Publ. Math. 92 (2000), 151–194.
  • [14] P.-E. Caprace and M. Sageev, Rank rigidity for $\mathrm{CAT}(0)$ cube complexes, Geom. Funct. Anal. 21 (2011), 851–891.
  • [15] R. Charney, An introduction to right-angled Artin groups, Geom. Dedicata 125 (2007), 141–158.
  • [16] R. Charney, Problems related to Artin groups, preprint, http://people.brandeis.edu/~charney/papers/Artin_probs.pdf.
  • [17] R. Charney, J. Crisp, and K. Vogtmann, Automorphisms of 2-dimensional right-angled Artin groups, Geom. Topol. 11 (2007), 2227–2264.
  • [18] R. Charney and M. W. Davis, “Finite $K(\pi,1)$s for Artin groups” in Prospects in Topology (Princeton, NJ, 1994), Ann. of Math. Stud. 138, Princeton Univ. Press, Princeton, 1995, 110–124.
  • [19] R. Charney and M. W. Davis, The $K(\pi,1)$-problem for hyperplane complements associated to infinite reflection groups, J. Amer. Math. Soc. 8 (1995), 597–627.
  • [20] R. Charney and M. Farber, Random groups arising as graph products, Algebr. Geom. Topol 12 (2012), 979–995.
  • [21] C. B. Croke and B. Kleiner, Spaces with nonpositive curvature and their ideal boundaries, Topology 39 (2000), 549–556.
  • [22] M. W. Davis, “Buildings are $\mathrm{CAT}(0)$” in Geometry and Cohomology in Group Theory (Durham, England, 1994), London Math. Soc. Lecture Note Ser. 252, Cambridge Univ. Press, Cambridge, 1998, 108–123.
  • [23] M. B. Day, Finiteness of outer automorphism groups of random right-angled Artin groups, Algebr. Geom. Topol. 12 (2012), 1553–1583.
  • [24] C. Droms, Isomorphisms of graph groups, Proc. Amer. Math. Soc. 100 (1987), 407–408.
  • [25] M. J. Dunwoody, The accessibility of finitely presented groups, Invent. Math. 81 (1985), 449–457.
  • [26] R. Gitik, M. Mitra, E. Rips, and M. Sageev, Widths of subgroups, Trans. Amer. Math. Soc. 350 (1998), 321–329.
  • [27] M. Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math. 53 (1981), 53–78.
  • [28] M. Gromov, “Hyperbolic manifolds, groups and actions” in Riemann Surfaces and Related Topics: Proceedings of the 1978Stony Brook Conference (Stony Brook, N.Y., 1978), Ann. of Math. Stud. 97, Princeton Univ. Press, Princeton, 1981, 183–213.
  • [29] M. Gromov, “Hyperbolic groups” in Essays in Group Theory, Math. Sci. Res. Inst. Publ. 8, Springer, New York, 1987, 75–263.
  • [30] M. F. Hagen, Cocompactly cubulated crystallographic groups, J. Lond. Math. Soc. (2) 90 (2014), 140–166.
  • [31] M. F. Hagen and P. Przytycki, Cocompactly cubulated graph manifolds, Israel J. Math. 207 (2015), 377–394.
  • [32] F. Haglund, Finite index subgroups of graph products, Geom. Dedicata 135 (2008), 167–209.
  • [33] F. Haglund and F. Paulin, Constructions arborescentes d’immeubles, Math. Ann. 325 (2003), 137–164.
  • [34] F. Haglund and D. T. Wise, Special cube complexes, Geom. Funct. Anal. 17 (2008), 1551–1620.
  • [35] G. C. Hruska and D. T. Wise, Finiteness properties of cubulated groups, Compos. Math. 150 (2014), 453–506.
  • [36] J. Huang, Quasi-isometric classification of right-angled Artin groups, I: The finite out case, Geom. Topol. 21 (2017), 3467–3537.
  • [37] J. Huang, Top-dimensional quasiflats in $\mathrm{CAT}(0)$ cube complexes, Geom. Topol. 21 (2017), 2281–2352.
  • [38] J. Huang, Commensurability of groups quasi-isometric to RAAGs, preprint, arXiv:1603.08586v2 [math.GT].
  • [39] T. Januszkiewicz and J. Świątkowski, Commensurability of graph products, Algebr. Geom. Topol. 1 (2001), 587–603.
  • [40] J. Kahn and V. Markovic, Immersing almost geodesic surfaces in a closed hyperbolic three manifold, Ann. of Math. (2) 175 (2012), 1127–1190.
  • [41] M. Kapovich, B. Kleiner, and B. Leeb, Quasi-isometries and the de Rham decomposition, Topology 37 (1998), 1193–1211.
  • [42] M. Kapovich and B. Leeb, Quasi-isometries preserve the geometric decomposition of Haken manifolds, Invent. Math. 128 (1997), 393–416.
  • [43] M. Kapovich and J. J. Millson, On representation varieties of Artin groups, projective arrangements and the fundamental groups of smooth complex algebraic varieties, Inst. Hautes Études Sci. Publ. Math. 88 (1998), 5–95.
  • [44] S.-H. Kim and T. Koberda, Embedability between right-angled Artin groups, Geom. Topol. 17 (2013), 493–530.
  • [45] B. Kleiner and B. Leeb, Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings, C. R. Acad. Sci. Paris Sér. I Math. 324 (1997), 639–643.
  • [46] B. Kleiner and B. Leeb, Groups quasi-isometric to symmetric spaces, Comm. Anal. Geom. 9 (2001), 239–260.
  • [47] M. R. Laurence, A generating set for the automorphism group of a graph group, J. Lond. Math. Soc. (2) 52 (1995), 318–334.
  • [48] B. Leeb, $3$-manifolds with(out) metrics of nonpositive curvature, Invent. Math. 122 (1995), 277–289.
  • [49] L. Mosher, M. Sageev, and K. Whyte, Quasi-actions on trees, I: Bounded valence, Ann. of Math. (2) 158 (2003), 115–164.
  • [50] P. Pansu, Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un, Ann. of Math. (2) 129 (1989), 1–60.
  • [51] P. Papasoglu and K. Whyte, Quasi-isometries between groups with infinitely many ends, Comment. Math. Helv. 77 (2002), 133–144.
  • [52] M. Ronan, Lectures on Buildings: Updated and Revised, Univ. Chicago Press, Chicago, 2009.
  • [53] M. Sageev, Ends of group pairs and non-positively curved cube complexes, Proc. Lond. Math. Soc. (3) 71 (1995), 585–617.
  • [54] M. Sageev, “CAT(0) cube complexes and groups” in Geometric Group Theory, IAS/Park City Math. Ser. 21, Amer. Math. Soc., Providence, 2014, 7–54.
  • [55] G. P. Scott and G. A. Swarup, An algebraic annulus theorem, Pacific J. Math. 196 (2000), 461–506.
  • [56] H. Servatius, Automorphisms of graph groups, J. Algebra 126 (1989), 34–60.
  • [57] J. R. Stallings, On torsion-free groups with infinitely many ends, Ann. of Math. (2) 88 (1968), 312–334.
  • [58] D. Sullivan, “On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions” in Riemann Surfaces and Related Topics: Proceedings of the 1978Stony Brook Conference (Stony Brook, N.Y., 1978), Ann. of Math. Stud. 97, Princeton Univ. Press, Princeton, 1981, 465–496.
  • [59] P. Tukia, On quasiconformal groups, J. Analyse Math. 46 (1986), 318–346.
  • [60] K. Whyte, Coarse bundles, preprint, arXiv:1006.3347v1 [math.GT].
  • [61] D. T. Wise, Non-positively curved squared complexes: Aperiodic tilings and non-residually finite groups, Ph.D. dissertation, Princeton University, Princeton, N.J., 1996.
  • [62] D. T. Wise, The structure of groups with a quasiconvex hierarchy, preprint, 2011.