The asymptotic geometry of right-angled Artin groups, I

Bestvina, Mladen¹ ; Kleiner, Bruce² ; Sageev, Michah³

¹ Department of Mathematics, University of Utah, 155 South 1400 East, Room 233, Salt Lake City, UT 84112-0090
² Yale University, Mathematics Department, PO Box 208283, New Haven, CT 06520-8283
³ Department of Mathematics, Technion – Israel Institute of Technology, Haifa 32000, Israel

Geometry & topology, Tome 12 (2008) no. 3, pp. 1653-1699.

Résumé

We study atomic right-angled Artin groups – those whose defining graph has no cycles of length $\le 4$, and no separating vertices, separating edges, or separating vertex stars. We show that these groups are not quasi-isometrically rigid, but that an intermediate form of rigidity does hold. We deduce from this that two atomic groups are quasi-isometric iff they are isomorphic.

Détail

DOI : 10.2140/gt.2008.12.1653

Keywords: CAT(0), quasi-isometry, rigidity

Bibliographie

[1] A R Ahlin, The large scale geometry of products of trees, Geom. Dedicata 92 (2002) 179

[2] H Bass, The degree of polynomial growth of finitely generated nilpotent groups, Proc. London Math. Soc. $(3)$ 25 (1972) 603

[3] J Behrstock, W Neumann, Quasi-isometric classification of graph manifold groups

[4] M. Bestvina, B. Kleiner, M. Sageev, Quasiflats in $\mathrm{CAT}(0)$ $2$–complexes

[5] M Burger, S Mozes, Lattices in product of trees, Inst. Hautes Études Sci. Publ. Math. (2000)

[6] R Charney, An introduction to right angled Artin groups

[7] R. Charney, J. Crisp, K. Vogtmann, Automorphisms of two-dimensional right-angled artin groups

[8] R Charney, M W Davis, The $K(\pi,1)$–problem for hyperplane complements associated to infinite reflection groups, J. Amer. Math. Soc. 8 (1995) 597

[9] M W Davis, Buildings are $\mathrm{CAT}(0)$, from: "Geometry and cohomology in group theory (Durham, 1994)", London Math. Soc. Lecture Note Ser. 252, Cambridge Univ. Press (1998) 108

[10] M W Davis, T Januszkiewicz, Right-angled Artin groups are commensurable with right-angled Coxeter groups, J. Pure Appl. Algebra 153 (2000) 229

[11] C Droms, Isomorphisms of graph groups, Proc. Amer. Math. Soc. 100 (1987) 407

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

[13] M Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes Études Sci. Publ. Math. (1981) 53

[14] M Gromov, Hyperbolic groups, from: "Essays in group theory", Math. Sci. Res. Inst. Publ. 8, Springer (1987) 75

[15] M Gromov, Asymptotic invariants of infinite groups, from: "Geometric group theory, Vol. 2 (Sussex, 1991)", London Math. Soc. Lecture Note Ser. 182, Cambridge Univ. Press (1993) 1

[16] J Harlander, H Meinert, Higher generation subgroup sets and the virtual cohomological dimension of graph products of finite groups, J. London Math. Soc. $(2)$ 53 (1996) 99

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

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

[19] A Karrass, A Pietrowski, D Solitar, Finite and infinite cyclic extensions of free groups, J. Austral. Math. Soc. 16 (1973) 458

[20] K H Kim, L Makar-Limanov, J Neggers, F W Roush, Graph algebras, J. Algebra 64 (1980) 46

[21] B Kleiner, B Leeb, Induced quasi-actions: a remark, preprint (2007)

[22] M R Laurence, A generating set for the automorphism group of a graph group, J. London Math. Soc. $(2)$ 52 (1995) 318

[23] J P Mccammond, D T Wise, Fans and ladders in small cancellation theory, Proc. London Math. Soc. $(3)$ 84 (2002) 599

[24] L Mosher, M Sageev, K Whyte, Quasi-actions on trees. I. Bounded valence, Ann. of Math. $(2)$ 158 (2003) 115

[25] M Sageev, Ends of group pairs and non-positively curved cube complexes, Proc. London Math. Soc. $(3)$ 71 (1995) 585

[26] H Servatius, Automorphisms of graph groups, J. Algebra 126 (1989) 34

[27] J R Stallings, On torsion-free groups with infinitely many ends, Ann. of Math. $(2)$ 88 (1968) 312

[28] D Wise, Non-positively curved squared complexes, aperiodic tilings, and non-residually finite groups, PhD thesis, Princeton (1996)