×

zbMATH — the first resource for mathematics

Groups quasi-isometric to right-angled Artin groups. (English) Zbl 1432.20030
Summary: 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 \(\operatorname{CAT}(0)\) 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.

MSC:
20F65 Geometric group theory
20F36 Braid groups; Artin groups
20F69 Asymptotic properties of groups
PDF BibTeX XML Cite
Full Text: DOI Euclid
References:
[1] P. Abramenko and K. S. Brown,Buildings: Theory and Applications, Grad. Texts in Math.248, Springer, New York, 2008. · Zbl 1214.20033
[2] I. Agol,The virtual Haken conjecture, Doc. Math.18(2013), 1045-1087. · Zbl 1286.57019
[3] A. R. Ahlin,The large scale geometry of products of trees, Geom. Dedicata92(2002), 179-184. · Zbl 1009.20033
[4] H. Bass,The degree of polynomial growth of finitely generated nilpotent groups, Proc. Lond. Math. Soc. (3)25(1972), 603-614. · Zbl 0259.20045
[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. · Zbl 1226.20033
[6] J. A. Behrstock and W. D. Neumann,Quasi-isometric classification of graph manifold groups, Duke Math. J.141(2008), 217-240. · Zbl 1194.20045
[7] N. Bergeron and D. T. Wise,A boundary criterion for cubulation, Amer. J. Math.134(2012), 843-859. · Zbl 1279.20051
[8] M. Bestvina and N. Brady,Morse theory and finiteness properties of groups, Invent. Math.129(1997), 445-470. · Zbl 0888.20021
[9] M. Bestvina, B. Kleiner, and M. Sageev,The asymptotic geometry of right-angled Artin groups, I, Geom. Topol.12(2008), 1653-1699. · Zbl 1203.20038
[10] M. Bestvina, B. Kleiner, and M. Sageev,Quasiflats in \(\text{CAT}(0)\) \(2\)-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. · Zbl 1029.20018
[12] M. R. Bridson and A. Haefliger,Metric Spaces of Non-Positive Curvature, Grundlehren Math. Wissen.319, Springer, Berlin, 1999. · Zbl 0988.53001
[13] M. Burger and S. Mozes,Lattices in product of trees, Inst. Hautes Études Sci. Publ. Math.92(2000), 151-194. · Zbl 1007.22013
[14] P.-E. Caprace and M. Sageev,Rank rigidity for \(\text{CAT}(0)\) cube complexes, Geom. Funct. Anal.21(2011), 851-891. · Zbl 1266.20054
[15] R. Charney,An introduction to right-angled Artin groups, Geom. Dedicata125(2007), 141-158. · Zbl 1152.20031
[16] R. Charney,Problems related to Artin groups, preprint,http://people.brandeis.edu/ charney/papers/Artin_probs.pdf. · Zbl 1152.20031
[17] R. Charney, J. Crisp, and K. Vogtmann,Automorphisms of 2-dimensional right-angled Artin groups, Geom. Topol.11(2007), 2227-2264. · Zbl 1152.20032
[18] R. Charney and M. W. Davis, “Finite \(K(π,1)\)s for Artin groups” inProspects in Topology (Princeton, NJ, 1994), Ann. of Math. Stud.138, Princeton Univ. Press, Princeton, 1995, 110-124. · Zbl 0930.55006
[19] R. Charney and M. W. Davis,The \(K(π,1)\)-problem for hyperplane complements associated to infinite reflection groups, J. Amer. Math. Soc.8(1995), 597-627. · Zbl 0833.51006
[20] R. Charney and M. Farber,Random groups arising as graph products, Algebr. Geom. Topol12(2012), 979-995. · Zbl 1280.20046
[21] C. B. Croke and B. Kleiner,Spaces with nonpositive curvature and their ideal boundaries, Topology39(2000), 549-556. · Zbl 0959.53014
[22] M. W. Davis, “Buildings are \(\text{CAT}(0)\)” inGeometry 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. · Zbl 1246.05141
[24] C. Droms,Isomorphisms of graph groups, Proc. Amer. Math. Soc.100(1987), 407-408. · Zbl 0619.20015
[25] M. J. Dunwoody,The accessibility of finitely presented groups, Invent. Math.81(1985), 449-457. · Zbl 0572.20025
[26] R. Gitik, M. Mitra, E. Rips, and M. Sageev,Widths of subgroups, Trans. Amer. Math. Soc.350(1998), 321-329. · Zbl 0897.20030
[27] M. Gromov,Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math.53(1981), 53-78. · Zbl 0474.20018
[28] M. Gromov, “Hyperbolic manifolds, groups and actions” inRiemann 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” inEssays 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. · Zbl 1342.20044
[31] M. F. Hagen and P. Przytycki,Cocompactly cubulated graph manifolds, Israel J. Math.207(2015), 377-394. · Zbl 1330.57030
[32] F. Haglund,Finite index subgroups of graph products, Geom. Dedicata135(2008), 167-209. · Zbl 1195.20047
[33] F. Haglund and F. Paulin,Constructions arborescentes d’immeubles, Math. Ann.325(2003), 137-164. · Zbl 1025.51014
[34] F. Haglund and D. T. Wise,Special cube complexes, Geom. Funct. Anal.17(2008), 1551-1620. · Zbl 1155.53025
[35] G. C. Hruska and D. T. Wise,Finiteness properties of cubulated groups, Compos. Math.150(2014), 453-506. · Zbl 1335.20043
[36] J. Huang,Quasi-isometric classification of right-angled Artin groups, I: The finite out case, Geom. Topol.21(2017), 3467-3537. · Zbl 1404.20033
[37] J. Huang,Top-dimensional quasiflats in \(\text{CAT}(0)\) cube complexes, Geom. Topol.21(2017), 2281-2352. · Zbl 1439.20045
[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. · Zbl 0998.20029
[40] J. Kahn and V. Markovic,Immersing almost geodesic surfaces in a closed hyperbolic three manifold, Ann. of Math. (2)175(2012), 1127-1190. · Zbl 1254.57014
[41] M. Kapovich, B. Kleiner, and B. Leeb,Quasi-isometries and the de Rham decomposition, Topology37(1998), 1193-1211. · Zbl 0954.53027
[42] M. Kapovich and B. Leeb,Quasi-isometries preserve the geometric decomposition of Haken manifolds, Invent. Math.128(1997), 393-416. · Zbl 0866.20033
[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. · Zbl 0982.20023
[44] S.-H. Kim and T. Koberda,Embedability between right-angled Artin groups, Geom. Topol.17(2013), 493-530. · Zbl 1278.20049
[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. · Zbl 0882.53037
[46] B. Kleiner and B. Leeb,Groups quasi-isometric to symmetric spaces, Comm. Anal. Geom.9(2001), 239-260. · Zbl 1035.53073
[47] M. R. Laurence,A generating set for the automorphism group of a graph group, J. Lond. Math. Soc. (2)52(1995), 318-334. · Zbl 0836.20036
[48] B. Leeb,\(3\)-manifolds with(out) metrics of nonpositive curvature, Invent. Math.122(1995), 277-289. · Zbl 0840.53031
[49] L. Mosher, M. Sageev, and K. Whyte,Quasi-actions on trees, I: Bounded valence, Ann. of Math. (2)158(2003), 115-164. · Zbl 1038.20016
[50] P. Pansu,Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un, Ann. of Math. (2)129(1989), 1-60. · Zbl 0678.53042
[51] P. Papasoglu and K. Whyte,Quasi-isometries between groups with infinitely many ends, Comment. Math. Helv.77(2002), 133-144. · Zbl 1010.20026
[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. · Zbl 0861.20041
[54] M. Sageev, “CAT(0) cube complexes and groups” inGeometric 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. · Zbl 0984.20028
[56] H. Servatius,Automorphisms of graph groups, J. Algebra126(1989), 34-60. · Zbl 0682.20022
[57] J. R. Stallings,On torsion-free groups with infinitely many ends, Ann. of Math. (2)88(1968), 312-334. · Zbl 0238.20036
[58] D. Sullivan, “On the ergodic theory at infinity of an arbitrary discrete group of hyperbolic motions” inRiemann 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. · Zbl 0603.30026
[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.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.