On the asymptotic dimension of the curve complex

Bestvina, Mladen¹ ; Bromberg, Ken¹

¹ Department of Mathematics, University of Utah, Salt Lake City, UT, United States

Geometry & topology, Tome 23 (2019) no. 5, pp. 2227-2276

Résumé

We give a bound, linear in the complexity of the surface, to the asymptotic dimension of the curve complex as well as the capacity dimension of the ending lamination space.

Détail

DOI : 10.2140/gt.2019.23.2227

Classification : 20F65
Keywords: asymptotic dimension, curve complex

Bibliographie

[1] I Agol, Ideal triangulations of pseudo-Anosov mapping tori, from: "Topology and geometry in dimension three" (editors W Li, L Bartolini, J Johnson, F Luo, R Myers, J H Rubinstein), Contemp. Math. 560, Amer. Math. Soc. (2011) 1 | DOI

[2] T Aougab, Quadratic bounds on the quasiconvexity of nested train track sequences, Topology Proc. 44 (2014) 365

[3] J Behrstock, M F Hagen, A Sisto, Asymptotic dimension and small-cancellation for hierarchically hyperbolic spaces and groups, Proc. Lond. Math. Soc. 114 (2017) 890 | DOI

[4] G C Bell, K Fujiwara, The asymptotic dimension of a curve graph is finite, J. Lond. Math. Soc. 77 (2008) 33 | DOI

[5] M Bestvina, K Bromberg, K Fujiwara, Constructing group actions on quasi-trees and applications to mapping class groups, Publ. Math. Inst. Hautes Études Sci. 122 (2015) 1 | DOI

[6] M Bonk, O Schramm, Embeddings of Gromov hyperbolic spaces, Geom. Funct. Anal. 10 (2000) 266 | DOI

[7] B H Bowditch, Tight geodesics in the curve complex, Invent. Math. 171 (2008) 281 | DOI

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

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

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

[11] D Gabai, On the topology of ending lamination space, Geom. Topol. 18 (2014) 2683 | DOI

[12] É Ghys, P De La Harpe, Sur les groupes hyperboliques d’après Mikhael Gromov, 83, Birkhäuser (1990) | DOI

[13] B Grünbaum, Convex polytopes, 221, Springer (2003) | DOI

[14] U Hamenstädt, Geometry of the mapping class groups, I : Boundary amenability, Invent. Math. 175 (2009) 545 | DOI

[15] J L Harer, The virtual cohomological dimension of the mapping class group of an orientable surface, Invent. Math. 84 (1986) 157 | DOI

[16] S Hensel, P Przytycki, The ending lamination space of the five-punctured sphere is the Nöbeling curve, J. Lond. Math. Soc. 84 (2011) 103 | DOI

[17] D Hume, Embedding mapping class groups into a finite product of trees, Groups Geom. Dyn. 11 (2017) 613 | DOI

[18] E Klarreich, The boundary at infinity of the curve complex and the relative Teichmüller space, preprint (2018)

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

[20] J M Mackay, A Sisto, Embedding relatively hyperbolic groups in products of trees, Algebr. Geom. Topol. 13 (2013) 2261 | DOI

[21] H A Masur, Y N Minsky, Geometry of the complex of curves, I : Hyperbolicity, Invent. Math. 138 (1999) 103 | DOI

[22] H A Masur, Y N Minsky, Geometry of the complex of curves, II : Hierarchical structure, Geom. Funct. Anal. 10 (2000) 902 | DOI

[23] H A Masur, Y N Minsky, Quasiconvexity in the curve complex, from: "In the tradition of Ahlfors and Bers, III" (editors W Abikoff, A Haas), Contemp. Math. 355, Amer. Math. Soc. (2004) 309 | DOI

[24] R C Penner, J L Harer, Combinatorics of train tracks, 125, Princeton Univ. Press (1992) | DOI

[25] M Sapir, A Higman embedding preserving asphericity, J. Amer. Math. Soc. 27 (2014) 1 | DOI

[26] W P Thurston, The geometry and topology of three-manifolds, lecture notes (1979)

[27] R C H Webb, Combinatorics of tight geodesics and stable lengths, Trans. Amer. Math. Soc. 367 (2015) 7323 | DOI

[28] G M Ziegler, Lectures on polytopes, 152, Springer (1995) | DOI