Embedding relatively hyperbolic groups in products of trees

Mackay, John M¹ ; Sisto, Alessandro¹

¹Mathematical Institute, University of Oxford, 24–29 St Giles’, Oxford OX1 3LB, UK

Algebraic and Geometric Topology, Tome 13 (2013) no. 4, pp. 2261-2282

Résumé

We show that a relatively hyperbolic group quasi-isometrically embeds in a product of finitely many trees if the peripheral subgroups do, and we provide an estimate on the minimal number of trees needed. Applying our result to the case of 3–manifolds, we show that fundamental groups of closed 3–manifolds have linearly controlled asymptotic dimension at most 8. To complement this result, we observe that fundamental groups of Haken 3–manifolds with non-empty boundary have asymptotic dimension 2.

Détail

DOI : 10.2140/agt.2013.13.2261

Classification : 20F65, 20F69
Keywords: relatively hyperbolic group, asymptotic Assouad–Nagata dimension, linearly controlled asymptotic dimension, product of trees

Bibliographie

[1] I Agol, Criteria for virtual fibering, J. Topol. 1 (2008) 269

[2] M Aschenbrenner, S Friedl, H Wilton, 3-manifold groups

[3] G Bell, A Dranishnikov, On asymptotic dimension of groups, Algebr. Geom. Topol. 1 (2001) 57

[4] G Bell, A Dranishnikov, On asymptotic dimension of groups acting on trees, Geom. Dedicata 103 (2004) 89

[5] G Bell, A Dranishnikov, A Hurewicz-type theorem for asymptotic dimension and applications to geometric group theory, Trans. Amer. Math. Soc. 358 (2006) 4749

[6] M Bestvina, K Bromberg, K Fujiwara, Constructing group actions on quasi-trees and applications to mapping class groups

[7] B H Bowditch, Relatively hyperbolic groups, Internat. J. Algebra Comput. 22 (2012) 1250016, 66

[8] N Brodskiy, J Dydak, J Higes, A Mitra, Assouad–Nagata dimension via Lipschitz extensions, Israel J. Math. 171 (2009) 405

[9] N Brodskiy, J Dydak, M Levin, A Mitra, A Hurewicz theorem for the Assouad–Nagata dimension, J. Lond. Math. Soc. 77 (2008) 741

[10] N Brodskiy, J Higes, Assouad–Nagata dimension of tree-graded spaces

[11] M Burger, A Iozzi, N Monod, Equivariant embeddings of trees into hyperbolic spaces, Int. Math. Res. Not. 2005 (2005) 1331

[12] S Buyalo, Asymptotic dimension of a hyperbolic space and the capacity dimension of its boundary at infinity, Algebra i Analiz 17 (2005) 70

[13] S Buyalo, Capacity dimension and embedding of hyperbolic spaces into the product of trees, Algebra i Analiz 17 (2005) 42

[14] S Buyalo, A Dranishnikov, V Schroeder, Embedding of hyperbolic groups into products of binary trees, Invent. Math. 169 (2007) 153

[15] S Buyalo, N Lebedeva, Dimensions of locally and asymptotically self-similar spaces, Algebra i Analiz 19 (2007) 60

[16] S Buyalo, V Schroeder, Embedding of hyperbolic spaces in the product of trees, Geom. Dedicata 113 (2005) 75

[17] S Buyalo, V Schroeder, Elements of asymptotic geometry, EMS Monographs in Mathematics, European Mathematical Society (EMS), Zürich (2007)

[18] S Buyalo, V Shrëder, The hyperbolic dimension of metric spaces, Algebra i Analiz 19 (2007) 93

[19] Y De Cornulier, Dimension of asymptotic cones of Lie groups, J. Topol. 1 (2008) 342

[20] F Dahmani, Combination of convergence groups, Geom. Topol. 7 (2003) 933

[21] A Dranishnikov, On asymptotic dimension of amalgamated products and right-angled Coxeter groups, Algebr. Geom. Topol. 8 (2008) 1281

[22] A Dranishnikov, Cohomological approach to asymptotic dimension, Geom. Dedicata 141 (2009) 59

[23] A Dranishnikov, J Smith, On asymptotic Assouad–Nagata dimension, Topology Appl. 154 (2007) 934

[24] C Druţu, M Sapir, Tree-graded spaces and asymptotic cones of groups, Topology 44 (2005) 959

[25] J Dydak, J Higes, Asymptotic cones and Assouad–Nagata dimension, Proc. Amer. Math. Soc. 136 (2008) 2225

[26] B Farb, Relatively hyperbolic groups, Geom. Funct. Anal. 8 (1998) 810

[27] Ś R Gal, Asymptotic dimension and uniform embeddings, Groups Geom. Dyn. 2 (2008) 63

[28] M Gromov, Asymptotic invariants of infinite groups, from: "Geometric group theory, Vol. 2" (editors G A Niblo, M A Roller), London Math. Soc. Lecture Note Ser. 182, Cambridge Univ. Press (1993) 1

[29] D Groves, J F Manning, Dehn filling in relatively hyperbolic groups, Israel J. Math. 168 (2008) 317

[30] J Higes, I Peng, Assouad–Nagata dimension of connected Lie groups, Math. Z. 273 (2013) 283

[31] D Hume, Direct embeddings of relatively hyperbolic groups with optimal $\ell^p$ compression exponent

[32] D Hume, Embedding mapping class groups into finite products of trees

[33] D Hume, A Sisto, Embedding universal covers of graph manifolds in products of trees, to appear in Proc. Amer. Math. Soc.

[34] I Kapovich, N Benakli, Boundaries of hyperbolic groups, from: "Combinatorial and geometric group theory" (editors S Cleary, R Gilman, A G Myasnikov, V Shpilrain), Contemp. Math. 296, Amer. Math. Soc. (2002) 39

[35] U Lang, T Schlichenmaier, Nagata dimension, quasisymmetric embeddings, and Lipschitz extensions, Int. Math. Res. Not. 2005 (2005) 3625

[36] Y Liu, Virtual cubulation of nonpositively curved graph manifolds

[37] J M Mackay, A Sisto, Quasi-hyperbolic planes in relatively hyperbolic groups

[38] P W Nowak, On exactness and isoperimetric profiles of discrete groups, J. Funct. Anal. 243 (2007) 323

[39] D Osin, Asymptotic dimension of relatively hyperbolic groups, Int. Math. Res. Not. 2005 (2005) 2143

[40] S D Pauls, The large scale geometry of nilpotent Lie groups, Comm. Anal. Geom. 9 (2001) 951

[41] P Przytycki, D T Wise, Mixed 3-manifolds are virtually special

[42] A Sisto, Contracting elements and random walks

[43] A Sisto, Projections and relative hyperbolicity, to appear in Enseign. Math. (2013)

[44] S Wang, F Yu, Graph manifolds with non-empty boundary are covered by surface bundles, Math. Proc. Cambridge Philos. Soc. 122 (1997) 447

[45] D Wise, The structure of groups with a quasiconvex hierarchy (2011)

[46] D T Wise, From riches to raags: 3-manifolds, right-angled Artin groups, and cubical geometry, CBMS Regional Conference Series in Mathematics 117, Published for the Conference Board of the Mathematical Sciences (2012)