Geodesic
Uniform hyperbolicity of the curve graph via surgery sequences
Clay, Matt1 ; Rafi, Kasra2 ; Schleimer, Saul3
1Department of Mathematical Sciences, University of Arkansas, SCEN 309, Fayetteville, AR 72701, USA
2Department of Mathematics, University of Toronto, Room 6290, 40 St. George Street, Toronto, Ontario M5S 2E4, Canada
3Mathematics Institute, University of Warwick, Zeeman Building, Conventry CV4 7AL, UK
Algebraic and Geometric Topology, Tome 14 (2014) no. 6, pp. 3325-3344
Cet article a éte moissonné depuis la source Mathematical Sciences Publishers

Voir la notice de l'article

We prove that the curve graph C(1)(S) is Gromov-hyperbolic with a constant of hyperbolicity independent of the surface S. The proof is based on the proof of hyperbolicity of the free splitting complex by Handel and Mosher, as interpreted by Hilion and Horbez.

DOI : 10.2140/agt.2014.14.3325
Classification : 57M99, 30F60
Keywords: curve complex, arc complex, Gromov hyperbolic

Clay, Matt  1   ; Rafi, Kasra  2   ; Schleimer, Saul  3

1 Department of Mathematical Sciences, University of Arkansas, SCEN 309, Fayetteville, AR 72701, USA
2 Department of Mathematics, University of Toronto, Room 6290, 40 St. George Street, Toronto, Ontario M5S 2E4, Canada
3 Mathematics Institute, University of Warwick, Zeeman Building, Conventry CV4 7AL, UK 
@article{10_2140_agt_2014_14_3325,
     author = {Clay, Matt and Rafi, Kasra and Schleimer, Saul},
     title = {Uniform hyperbolicity of the curve graph via surgery sequences},
     journal = {Algebraic and Geometric Topology},
     pages = {3325--3344},
     year = {2014},
     volume = {14},
     number = {6},
     doi = {10.2140/agt.2014.14.3325},
     url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2014.14.3325/}
}
TY  - JOUR
AU  - Clay, Matt
AU  - Rafi, Kasra
AU  - Schleimer, Saul
TI  - Uniform hyperbolicity of the curve graph via surgery sequences
JO  - Algebraic and Geometric Topology
PY  - 2014
SP  - 3325
EP  - 3344
VL  - 14
IS  - 6
UR  - http://geodesic.mathdoc.fr/articles/10.2140/agt.2014.14.3325/
DO  - 10.2140/agt.2014.14.3325
ID  - 10_2140_agt_2014_14_3325
ER  - 
%0 Journal Article
%A Clay, Matt
%A Rafi, Kasra
%A Schleimer, Saul
%T Uniform hyperbolicity of the curve graph via surgery sequences
%J Algebraic and Geometric Topology
%D 2014
%P 3325-3344
%V 14
%N 6
%U http://geodesic.mathdoc.fr/articles/10.2140/agt.2014.14.3325/
%R 10.2140/agt.2014.14.3325
%F 10_2140_agt_2014_14_3325
Clay, Matt; Rafi, Kasra; Schleimer, Saul. Uniform hyperbolicity of the curve graph via surgery sequences. Algebraic and Geometric Topology, Tome 14 (2014) no. 6, pp. 3325-3344. doi: 10.2140/agt.2014.14.3325

[1] T Aougab, Uniform hyperbolicity of the graphs of curves, Geom. Topol. 17 (2013) 2855

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

[3] M Bestvina, M Feighn, Hyperbolicity of the complex of free factors, Adv. Math. 256 (2014) 104

[4] B H Bowditch, Intersection numbers and the hyperbolicity of the curve complex, J. Reine Angew. Math. 598 (2006) 105

[5] B H Bowditch, Uniform hyperbolicity of the curve graphs, Pacific J. Math. 269 (2014) 269

[6] J F Brock, R D Canary, Y N Minsky, The classification of Kleinian surface groups, II: The ending lamination conjecture, Ann. of Math. 176 (2012) 1

[7] J F Brock, H A Masur, Y N Minsky, Asymptotics of Weil–Petersson geodesic, I: Ending laminations, recurrence and flows, Geom. Funct. Anal. 19 (2010) 1229

[8] U Hamenstädt, Geometry of the complex of curves and of Teichmüller space, from: "Handbook of Teichmüller theory, Vol. I" (editor A Papadopoulos), IRMA Lect. Math. Theor. Phys. 11, Eur. Math. Soc. (2007) 447

[9] M Handel, L Mosher, The free splitting complex of a free group, I: Hyperbolicity, Geom. Topol. 17 (2013) 1581

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

[11] A Hatcher, On triangulations of surfaces, Topology Appl. 40 (1991) 189

[12] S Hensel, P Przytycki, R C H Webb, Slim unicorns and uniform hyperbolicity for arc graphs and curve graphs,

[13] A Hilion, C Horbez, The hyperbolicity of the sphere complex via surgery paths,

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

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

[16] Y N Minsky, The classification of punctured-torus groups, Ann. of Math. 149 (1999) 559

[17] Y N Minsky, The classification of Kleinian surface groups, I: Models and bounds, Ann. of Math. 171 (2010) 1

[18] K Rafi, Hyperbolicity in Teichmüller space,

[19] K Rafi, A characterization of short curves of a Teichmüller geodesic, Geom. Topol. 9 (2005) 179

[20] K Rafi, S Schleimer, Covers and the curve complex, Geom. Topol. 13 (2009) 2141

Cité par Sources :