Top-dimensional quasiflats in CAT(0) cube complexes

Huang, Jingyin¹

¹ The Department of Mathematics and Statistics, McGill University, Burnside Hall, Room 1242, 805 Sherbrooke W., Montreal QC H3A 0B9, Canada

Geometry & topology, Tome 21 (2017) no. 4, pp. 2281-2352.

Résumé

We show that every $n$–quasiflat in an $n$–dimensional $CAT\left(0\right)$ cube complex is at finite Hausdorff distance from a finite union of $n$–dimensional orthants. Then we introduce a class of cube complexes, called weakly special cube complexes, and show that quasi-isometries between their universal covers preserve top-dimensional flats. This is the foundational result towards the quasi-isometric classification of right-angled Artin groups with finite outer automorphism group.

Some of our arguments also extend to $CAT\left(0\right)$ spaces of finite geometric dimension. In particular, we give a short proof of the fact that a top-dimensional quasiflat in a Euclidean building is Hausdorff close to a finite union of Weyl cones, which was previously established by Kleiner and Leeb (1997), Eskin and Farb (1997) and Wortman (2006) by different methods.

Détail

DOI : 10.2140/gt.2017.21.2281

Classification : 20F67, 20F65, 20F69
Keywords: quasiflats, CAT(0) cube complexes, weakly special cube complexes

Bibliographie

[1] A Abrams, N Brady, P Dani, M Duchin, R Young, Pushing fillings in right-angled Artin groups, J. Lond. Math. Soc. 87 (2013) 663 | DOI

[2] J Behrstock, R Charney, Divergence and quasimorphisms of right-angled Artin groups, Math. Ann. 352 (2012) 339 | DOI

[3] J Behrstock, M F Hagen, Cubulated groups: thickness, relative hyperbolicity, and simplicial boundaries, Groups Geom. Dyn. 10 (2016) 649 | DOI

[4] J Behrstock, M F Hagen, A Sisto, Hierarchically hyperbolic spaces, I: Curve complexes for cubical groups, preprint (2014)

[5] J A Behrstock, T Januszkiewicz, W D Neumann, Quasi-isometric classification of some high dimensional right-angled Artin groups, Groups Geom. Dyn. 4 (2010) 681 | DOI

[6] J Behrstock, B Kleiner, Y Minsky, L Mosher, Geometry and rigidity of mapping class groups, Geom. Topol. 16 (2012) 781 | DOI

[7] J A Behrstock, W D Neumann, Quasi-isometric classification of graph manifold groups, Duke Math. J. 141 (2008) 217 | DOI

[8] M Bestvina, B Kleiner, M Sageev, The asymptotic geometry of right-angled Artin groups, I, Geom. Topol. 12 (2008) 1653 | DOI

[9] M Bestvina, B Kleiner, M Sageev, Quasiflats in CAT(0) 2–complexes, Algebr. Geom. Topol. 16 (2016) 2663 | DOI

[10] M R Bridson, Geodesics and curvature in metric simplicial complexes, PhD thesis, Cornell Univ. (1991)

[11] M R Bridson, A Haefliger, Metric spaces of non-positive curvature, 319, Springer (1999) | DOI

[12] P E Caprace, M Sageev, Rank rigidity for CAT(0) cube complexes, Geom. Funct. Anal. 21 (2011) 851 | DOI

[13] R Charney, An introduction to right-angled Artin groups, Geom. Dedicata 125 (2007) 141 | DOI

[14] P Dani, A Thomas, Divergence in right-angled Coxeter groups, Trans. Amer. Math. Soc. 367 (2015) 3549 | DOI

[15] M W Davis, G Moussong, Notes on nonpositively curved polyhedra, from: "Low dimensional topology" (editors K Böröczky Jr., W Neumann, A Stipsicz), Bolyai Soc. Math. Stud. 8, János Bolyai Math. Soc. (1999) 11

[16] M J Dunwoody, The accessibility of finitely presented groups, Invent. Math. 81 (1985) 449 | DOI

[17] A Eskin, B Farb, Quasi-flats and rigidity in higher rank symmetric spaces, J. Amer. Math. Soc. 10 (1997) 653 | DOI

[18] R Frigerio, J F Lafont, A Sisto, Rigidity of high dimensional graph manifolds, 372, Soc. Math. France (2015)

[19] M Gromov, Hyperbolic groups, from: "Essays in group theory" (editor S M Gersten), Math. Sci. Res. Inst. Publ. 8, Springer (1987) 75 | DOI

[20] F Haglund, D T Wise, Special cube complexes, Geom. Funct. Anal. 17 (2008) 1551 | DOI

[21] J Huang, Quasi-isometric classification of right-angled Artin groups, I: The finite out case, preprint (2014)

[22] J Huang, Quasi-isometry of right-angled artin groups, II : Several infinite out case, preprint (2015)

[23] J Huang, Commensurability of groups quasi-isometric to RAAG’s, preprint (2016)

[24] J Huang, B Kleiner, Groups quasi-isometric to RAAG’s, preprint (2016)

[25] W Hurewicz, H Wallman, Dimension theory, 4, Princeton Univ. Press (1941)

[26] M Kapovich, B Kleiner, B Leeb, Quasi-isometries and the de Rham decomposition, Topology 37 (1998) 1193 | DOI

[27] M Kapovich, B Leeb, Quasi-isometries preserve the geometric decomposition of Haken manifolds, Invent. Math. 128 (1997) 393 | DOI

[28] B Kleiner, The local structure of length spaces with curvature bounded above, Math. Z. 231 (1999) 409 | DOI

[29] B Kleiner, U Lang, Quasi-minimizing varieties in spaces of higher asymptotic rank, in preparation

[30] B Kleiner, B Leeb, Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings, C. R. Acad. Sci. Paris Sér. I Math. 324 (1997) 639 | DOI

[31] L Kramer, On the local structure and the homology of CAT(κ) spaces and Euclidean buildings, Adv. Geom. 11 (2011) 347 | DOI

[32] L Kramer, R M Weiss, Coarse rigidity of Euclidean buildings, preprint (2009)

[33] A Lytchak, K Nagano, Geodesically complete spaces with an upper curvature bound, in preparation

[34] G D Mostow, Strong rigidity of locally symmetric spaces, 78, Princeton Univ. Press (1973)

[35] K Nagano, A volume convergence theorem for Alexandrov spaces with curvature bounded above, Math. Z. 241 (2002) 127 | DOI

[36] M Sageev, Ends of group pairs and non-positively curved cube complexes, Proc. London Math. Soc. 71 (1995) 585 | DOI

[37] K Wortman, Quasiflats with holes in reductive groups, Algebr. Geom. Topol. 6 (2006) 91 | DOI