Median structures on asymptotic cones and homomorphisms into mapping class groups

Proceedings of the London Mathematical Society

Volumn 102, Issue 3, 2011, Pages 503-554

  Behrstock, Jason ^a   Druţu, Cornelia ^b   Sapir, Mark ^c  

^a CITY UNIVERSITY OF NEW YORK   (United States)

^b UNIVERSITY OF OXFORD   (United Kingdom)

^c VANDERBILT UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 79952417185     PISSN: 00246115     EISSN: 1460244X     Source Type: Journal    
DOI: 10.1112/plms/pdq025     Document Type: Article

References (57)

1. J. E. ANDERSEN, 'Mapping class groups do not have Kazhdan's property (T) ', Preprint, 2007, arXiv:0706.2184.
2. J. W. ANDERSON, J. ARAMAYONA and K. J. SHACKLETON, 'An obstruction to the strong relative hyperbolicity of a group', J. Group Theory 10 (2007) 749-756.
3. J. ARAMAYONA, C. LEININGER and J. SOUTO, 'Injections of mapping class groups,' Geom. Topol. 13 (2009) no. 5, 2523-2541.
4. J. A. BEHRSTOCK, 'Asymptotic geometry of the mapping class group and Teichmüller space', PhD Thesis, SUNY at Stony Brook, 2004, http://www.math.columbia.edu/~jason.
5. J. A. BEHRSTOCK, 'Asymptotic geometry of the mapping class group and Teichmüller space', Geom. Topol. 10 (2006) 1523-1578.
6. J. BEHRSTOCK, C. DRUŢU and L. MOSHER, 'Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity', Math. Ann. 344 (2009) 543-595.
7. J. BEHRSTOCK, B. KLEINER, Y. MINSKY and L. MOSHER, 'Geometry and rigidity of mapping class groups', Preprint, 2008, arXiv:0801.2006.
8. J. BEHRSTOCK and Y. MINSKY, 'Dimension and rank for mapping class groups', Ann. of Math. 167 (2008) 1055-1077.
9. I. BELEGRADEK and A. SZCZEPAŃSKI, 'Endomorphisms of relatively hyperbolic groups', Internat. J. Algebra Comput. 18 (2008) 97-110 (with an appendix by Oleg V. Belegradek).
10. M. BESTVINA, 'Degenerations of the hyperbolic space', Duke Math. J. 56 (1988) 143-161.
11. M. BESTVINA, 'R-trees in topology, geometry, and group theory', Handbook of geometric topology, Advanced Studies in Pure Mathematics (North-Holland, Amsterdam, 2002) 55-91.
12. J. BIRMAN, Braids, links, and mapping class groups, Annals of Mathematics Studies 82 (Princeton University Press, Princeton, 1974).
13. J. BIRMAN, A. LUBOTZKY and J. MCCARTHY, 'Abelian and solvable subgroups of the mapping class groups', Duke Math. J. 50 (1983) 1107-1120.
14. N. BOURBAKI, Topologie générale (Hermann, Paris, 1965).
15. B. BOWDITCH, 'One-ended subgroups of mapping class groups', Preprint, University of Warwick, 2007, http://www.warwick.ac.uk/masgak/preprints.html.
16. B. BOWDITCH, 'Tight geodesics in the curve complex', Invent. Math. 171 (2008) 281-300.
17. B. BOWDITCH, 'Atoroidal surface bundles over surfaces', Geom. Funct. Anal. 19 (2009) 943-988.
18. S. BURRIS and H. P. SANKAPPANAVAR, A course in universal algebra, Graduate Texts in Mathematics 83 (Springer, New York, 1981).
19. I. CHATTERJI, C. DRUŢU and F. HAGLUND, 'Kazhdan and Haagerup properties from the median viewpoint', Adv. Math., 225 (2010), 882-921.
20. J. CRISP and L. PARIS, 'The solution to a conjecture of Tits on the subgroup generated by the squares of the generators of an Artin group', Invent. Math. 145 (2001) 19-36.
21. F. DAHMANI and K. FUJIWARA, 'Copies of a one-ended group in a Mapping Class Group', Groups Geom. Dyn. 3 (2009) 359-377.
22. T. DELZANT and L. POTYAGAILO, 'Endomorphisms of Kleinian groups', Geom. Funct. Anal. 13 (2003) 396-436. (Pubitemid 36761286)
23. C. DRUŢU, 'Quasi-isometric classification of non-uniform lattices in semisimple groups of higher rank', Geom. Funct. Anal. 10 (2000) 327-388.
24. C. DRUŢU, 'Relatively hyperbolic groups: geometry and quasi-isometric invariance', Comment. Math. Helv. 84 (2009) 503-546.
25. C. DRUŢU, SH. MOZES and M. SAPIR, 'Groups acting on tree-graded spaces and splittings of relatively hyperbolic groups', Adv. Math. 217 (2007) 1313-1367.
26. C. DRUŢU, SH. MOZES and M. SAPIR, 'Divergence in lattices in semisimple lie groups and graphs of groups', Trans. Amer. Math. Soc. 362 (2010) 2451-2505.
27. C. DRUŢU and M. SAPIR, 'Tree-graded spaces and asymptotic cones of groups', Topology 44 (2005) 959-1058 (with an appendix by D. Osin and M. Sapir). (Pubitemid 40975221)
28. J. DUGUNDJI, Topology (Allyn and Bacon, Boston, 1966).
29. R. ENGELKING, Theory of dimensions finite and infinite, Sigma Series in Pure Mathematics 10 (Heldermann, Lemgo, 1995).
30. A. ESKIN, 'Quasi-isometric rigidity of nonuniform lattices in higher rank symmetric spaces', J. Amer. Math. Soc. 11 (1998) 321-361. (Pubitemid 128470380)
31. A. ESKIN and B. FARB, 'Quasi-flats and rigidity in higher rank symmetric spaces', J. Amer. Math. Soc. 10 (1997) 653-692.
32. B. FARB, 'Relatively hyperbolic groups', Geom. Funct. Anal. 8 (1998) 810-840.
33. B. FARB, A. LUBOTZKY and Y. MINSKY, 'Rank one phenomena for mapping class groups', Duke Math. J. 106 (2001) 581-597. (Pubitemid 33796901)
34. D. GROVES, 'Limits of (certain) CAT (0) groups, II: the Hopf property and the shortening argument', Preprint, 2004, arXiv:math. GR/0408080.
35. D. GROVES, 'Limit groups for relatively hyperbolic groups. II. Makanin-Razborov diagrams', Geom. Topol. 9 (2005) 2319-2358 (electronic).
36. D. GROVES, 'Limit groups for relatively hyperbolic groups, I: the basic tools', Algebr. Geom. Topol. 9 (2009) 1423-1466.
37. V. GUIRARDEL, 'Actions of finitely generated groups on R-trees', Ann. Inst. Fourier (Grenoble) 58 (2008) 159-211.
38. U. HAMENSTÄDT, 'Geometry of the mapping class groups III: quasi-isometric rigidity', Preprint, 2005, arXiv:math/0512429.
39. P. de la HARPE and A. VALETTE, 'La propriété (T) de Kazhdan pour les groupes localement compacts', Astérisque 175 (SociétéMathématique de France, Paris, 1989).
40. N. V. IVANOV, Subgroups of Teichmüller modular groups, Translations of Mathematical Monographs 115 (American Mathematical Society, Providence, RI, 1992).
41. N. V. IVANOV, 'Mapping class groups', Handbook of geometric topology (North-Holland, Amsterdam, 2002) 523-633.
42. B. KLEINER and B. LEEB, 'Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings', Publ. Math. Inst. Hautes Études Sci. 86 (1997) 115-197.
43. H. MASUR and Y. MINSKY, 'Geometry of the complex of curves, I. Hyperbolicity', Invent. Math. 138 (1999) 103-149.
44. H. MASUR and Y. MINSKY, 'Geometry of the complex of curves II: hierarchical structure', Geom. Funct. Anal. 10 (2000) 902-974.
45. Y. MINSKY, 'Curve complexes, surfaces and 3-manifolds', Proceedings of the International Congress of Mathematicians, vol. II (European Mathematical Society, Zürich, 2006) 1001-1033.
46. Y. MINSKY, 'The classification of Kleinian surface groups I: models and bounds', Ann. of Math. (2) 171 (2010) 1-107.
47. J. NAGATA, Modern dimension theory (Heldermann, Berlin, 1983).
48. B. H. NEUMANN, 'Groups with finite classes of conjugate elements', Proc. London Math. Soc. 1 (1951) 178-187.
49. G. A. NIBLO and L. D. REEVES, 'Groups acting on CAT (0) cube complexes', Geom. Topol. 1 (1997) 7.
50. G. A. NIBLO and M. ROLLER, 'Groups acting on cubes and Kazhdan's property (T) ', Proc. Amer. Math. Soc. 126 (1998) 693-699. (Pubitemid 128718632)
51. B. NICA, 'Group actions on median spaces', Preprint, 2008, arXiv:0809.4099.
52. K. OHSHIKA and L. POTYAGAILO, 'Self-embeddings of Kleinian groups', Ann. Sci. École Norm. Sup. 31 (1998) 329-343. (Pubitemid 128180577)
53. F. PAULIN, 'Topologie de Gromovéquivariante, structures hyperboliques et arbres réels', Invent. Math. 94 (1988) 53-80.
54. M. ROLLER, 'Poc sets, median algebras and group actions', Preprint, 1998, University of Southampton, http://www.maths.soton. ac.uk/pure/preprints.phtml.
55. R. SCHWARTZ, 'Quasi-isometric rigidity and Diophantine approximation', Acta Math. 177 (1996) 75-112. (Pubitemid 126343624)
56. L. VAN DEN DRIES and A. WILKIE, 'On Gromov's theorem concerning groups of polynomial growth and elementary logic', J. Algebra 89 (1984) 349-374.
57. E. R. VERHEUL, Multimedians in metric and normed spaces, CWI Tract 91 (Stichting Mathematisch Centrum voor Wiskunde en Informatica, Amsterdam, 1993).