Geometry and rigidity of mapping class groups

Geometry and Topology

Volumn 16, Issue 2, 2012, Pages 781-888

  Behrstock, Jason ^a   Kleiner, Bruce ^b   Minsky, Yair ^c   Mosher, Lee ^d  

^a CITY UNIVERSITY OF NEW YORK   (United States)

^b NEW YORK UNIVERSITY   (United States)

^c YALE UNIVERSITY   (United States)

^d RUTGERS UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 84863485944     PISSN: 14653060     EISSN: 13640380     Source Type: Journal    
DOI: 10.2140/gt.2012.16.781     Document Type: Article

References (59)

1. JE Andersen, MappingclassgroupsdonothaveKazhdan'sproperty(T) arXiv:0706.2184
2. J Behrstock, Asymptotic geometry of the mapping class group and Teichmüllerspace, PhD thesis, SUNY at Stony Brook (2004) Available at http://www.math.columbia.edu/~jason/thesis.pdf
3. J Behrstock, Asymptotic geometry of the mapping class group and Teichmüller space,Geom. Topol. 10 (2006) 1523-1578 MR2255505
4. J Behrstock, C Drutu, L Mosher, Thick metric spaces, relative hyperbolicity, andquasi-isometric rigidity, Math. Ann. 344 (2009) 543-595 MR2501302
5. J Behrstock, C Drutu, M Sapir, Median structures on asymptotic cones and homo-morphisms into mapping class groups, Proc. Lond. Math. Soc. 102 (2011) 503-554 MR2783135
6. J Behrstock, C Drutu, M Sapir, Addendum: Median structures on asymptotic conesand homomorphisms into mapping class groups. Proc. Lond. Math. Soc. 102 (2011)555-562 MR2783136
7. J Behrstock, Y N Minsky, Centroids and the rapid decay property in mapping classgroups, To appear in Jour. London Math. Soc. arXiv:0810.1969
8. J Behrstock, Y N Minsky, Dimension and rank for mapping class groups, Ann. ofMath. 167 (2008) 1055-1077 MR2415393
9. M Bourdon, H Pajot, Rigidity of quasi-isometries for some hyperbolic buildings,Comment. Math. Helv. 75 (2000) 701-736 MR1789183
10. JW Cannon, D Cooper, A characterization of cocompact hyperbolic and finite-volume hyperbolic groups in dimension three, Trans. Amer. Math. Soc. 330 (1992) 419-431 MR1036000
11. A Casson, D Jungreis, Convergence groups and Seifert fibered 3-manifolds, Invent.Math. 118 (1994) 441-456 MR1296353
12. C Drutu, M Sapir, Tree-graded spaces and asymptotic cones of groups, Topology 44(2005) 959-1058 MR2153979 With an appendix by D Osin and Sapir
13. T Dymarz, Large scale geometry of certain solvable groups, Geom. Funct. Anal. 19(2010)1650-1687 MR2594617
14. A Eskin, Quasi-isometric rigidity of nonuniform lattices in higher rank symmetricspaces, J. Amer. Math. Soc. 11 (1998) 321-361 MR1475886
15. A Eskin, B Farb, Quasi-flats and rigidity in higher rank symmetric spaces, J. Amer.Math. Soc. 10 (1997) 653-692 MR1434399
16. A Eskin, D Fisher, K Whyte, Quasi-isometries and rigidity of solvable groups, PureAppl. Math. Q. 3 (2007) 927-947 MR2402598
17. B Farb, A Lubotzky, Y Minsky, Rank-1 phenomena for mapping class groups, DukeMath. J. 106 (2001) 581-597 MR1813237
18. B Farb, H Masur, Superrigidity and mapping class groups, Topology 37 (1998) 1169-1176 MR1632912
19. B Farb, L Mosher, A rigidity theorem for the solvable Baumslag-Solitar groups,Invent. Math. 131 (1998) 419-451 MR1608595 With an appendix by D Cooper
20. B Farb, L Mosher, Quasi-isometric rigidity for the solvable Baumslag-Solitargroups, II, Invent. Math. 137 (1999) 613-649 MR1709862
21. B Farb, R Schwartz, The large-scale geometry of Hilbert modular groups, J. Differen-tial Geom. 44 (1996) 435-478 MR1431001
22. D Gabai, Convergence groups are Fuchsian groups, Ann. of Math. 136 (1992) 447-510 MR1189862
23. M Gromov, Groups of polynomial growth and expanding maps, Inst. Hautes ÉtudesSci. Publ. Math. (1981) 53-73 MR623534
24. M Gromov, Infinite groups as geometric objects, from: "Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983)", PWN, Warsaw (1984)385-392 MR804694
25. M Gromov, Asymptotic invariants of infinite groups, from: Geometric group theory, Vol. 2 (Sussex, 1991), (G A Niblo, M A Roller, editors), London Math. Soc. LectureNote Ser. 182, Cambridge Univ. Press (1993) 1-295 MR1253544
26. M Gromov, P Pansu, Rigidity of lattices: an introduction, from: Geometric topol-ogy: recent developments (Montecatini Terme, 1990), (P De Bartolomeis, F Tricerri,editors), Lecture Notes in Math. 1504, Springer, Berlin (1991) 39-137 MR1168043
27. U Hamenstädt, Geometry of the mapping class groups III: Quasi-isometric rigidityar Xiv:math.GT/0512429
28. JL Harer, Stability of the homology of the mapping class groups of orientable surfaces, Ann. of Math. 121 (1985) 215-249 MR786348
29. WJ Harvey, Boundary structure of the modular group, from: "Riemannsurfacesandrelated topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, 1978)", (I Kra, B Maskit, editors), Ann. of Math. Stud. 97, Princeton Univ. Press (1981)245-251 MR624817
30. A Hinkkanen, The structure of certain quasisymmetric groups, Mem. Amer. Math.Soc. 83, no. 422, Amer. Math. Soc. (1990) MR948926
31. NV Ivanov, Algebraic properties of the Teichmüller modular group, Dokl. Akad. Nauk SSSR 275 (1984) 786-789 MR745513 In Russian
32. [] NV Ivanov, Complexes of curves and Teichmüller modular groups, Uspekhi Mat. Nauk 42 (1987)49-91 MR896878 In Russian
33. NV Ivanov, Teichmüller modular groups and arithmetic groups, Zap. Nauchn.Sem. Leningrad. Otdel. Mat. Inst. Steklov.LOMI/ 167 (1988) 95-110, 190-191MR964258 In Russian
34. NV Ivanov, Automorphism of complexes of curves and of Teichmüller spaces, Internat. Math. Res. Notices (1997) 651-666 MR1460387
35. M Kapovich, Hyperbolic manifolds and discrete groups, Progress in Math. 183,Birkhäuser, Boston, MA (2001) MR1792613
36. B Kleiner, B Leeb, Rigidity of quasi-isometries for symmetric spaces and Euclideanbuildings, Inst. Hautes Études Sci. Publ. Math. (1997) 115-197 MR1608566
37. M Korkmaz, Automorphisms of complexes of curves on punctured spheres and onpunctured tori, Topology Appl. 95 (1999) 85-111 MR1696431
38. F Luo, Automorphisms of the complex of curves, Topology 39 (2000) 283-298 MR1722024
39. HA Masur, YN Minsky, Geometry of the complex of curves I: Hyperbolicity, Invent. Math. 138 (1999) 103-149 MR1714338
40. HA Masur, YN Minsky, Geometry of the complex of curves II: Hierarchical structure, Geom. Funct. Anal. 10 (2000) 902-974 MR1791145
41. J McCarthy, A Tits-alternative for subgroups of surface mapping class groups,Trans. Amer. Math. Soc. 291 (1985) 583-612 MR800253
42. J Milnor, Growth of finitely generated solvable groups, J. Differential Geometry 2(1968)447-449 MR0244899
43. J Milnor, A note on curvature and fundamental group, J. Differential Geometry 2(1968)1-7 MR0232311
44. L Mosher, Mapping class groups are automatic, Ann. of Math. 142 (1995) 303-384 MR1343324
45. L Mosher, Homology and dynamics in quasi-isometric rigidity of once-puncturedmapping class groups, from: Geometric and cohomological methods in group theory,(M R Bridson, P H Kropholler, I J Leary, editors), London Math. Soc. Lecture Note Ser.358, Cambridge Univ. Press (2009) 225-255 MR2605178
46. L Mosher, M Sageev, K Whyte, Quasi-actions on trees I. Bounded valence, Ann. ofMath. 158 (2003) 115-164 MR1998479
47. L Mosher, M Sageev, K Whyte, Quasi-actions on trees II: Finite depth Bass-Serretrees, Mem. Amer. Math. Soc. 214, no. 1008, Amer. Math. Soc. (2011) MR2867450
48. GD Mostow, Strong rigidity of locally symmetric spaces, Annals of Math. Studies78,Princeton Univ. Press (1973) MR0385004
49. P Pansu, Métriques de Carnot-Carathéodory et quasiisométries des espacessymétriques de rang un, Ann. of Math. 129 (1989) 1-60 MR979599
50. P Papasoglu, Quasi-isometry invariance of group splittings, Ann. of Math. 161 (2005)759-830 MR2153400
51. I Peng, Coarse differentiation and quasi-isometries of a class of solvable Lie groups I,Geom. Topol. 15 (2011) 1883-1925 MR2860983
52. RE Schwartz, The quasi-isometry classification of rank one lattices, Inst. HautesÉtudes Sci. Publ. Math. (1995) 133-168 MR1383215
53. JR Stallings, Ontorsion-free groups with infinitely many ends, Ann.of Math.88(1968)312-334 MR0228573
54. D Sullivan, On the ergodic theory at infinity of an arbitrary discrete group of hyperbolicmotions, from: Riemann surfaces and related topics: Proceedings of the 1978 StonyBrook Conference (State Univ. New York, 1978), (I Kra, B Maskit, editors), Ann. Of Math. Stud. 97, Princeton Univ. Press (1981) 465-496 MR624833
55. AS Švarc, A volume invariant of coverings, Dokl. Akad. Nauk SSSR 105 (1955) 32-34 MR0075634 In Russian
56. P Tukia, Quasiconformal extension of quasisymmetric mappings compatible with aMöbius group, Acta Math. 154 (1985) 153-193 MR781586
57. L van den Dries, A J Wilkie, Gromov's theorem on groups of polynomial growth andelementary logic, J. Algebra 89 (1984) 349-374 MR751150
58. K Whyte, The large scale geometry of the higher Baumslag-Solitar groups, Geom.Funct. Anal. 11 (2001) 1327-1343 MR1878322
59. X Xie, Quasi-isometric rigidity of Fuchsian buildings, Topology 45 (2006) 101-169 MR2170496