Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-8448b6f56d-dnltx Total loading time: 0 Render date: 2024-04-25T00:30:14.137Z Has data issue: false hasContentIssue false

2 - Topology of open nonpositively curved manifolds

Published online by Cambridge University Press:  05 January 2016

By
Igor Belegradek
Edited by
C. S. Aravinda ,
F. T. Farrell  and
J. -F. Lafont
Igor Belegradek
Affiliation:
Georgia Institute of Technology
C. S. Aravinda
Affiliation:
TIFR Centre for Applicable Mathematics, Bangalore, India
F. T. Farrell
Affiliation:
Tsinghua University, Beijing
J. -F. Lafont
Affiliation:
Ohio State University
Get access

Summary

Abstract

This is a survey of topological properties of open, complete nonpositively curved manifolds which may have infinite volume. Topics include topology of ends, restrictions on the fundamental group, as well as a review of known examples.

Among many monographs and surveys on aspects of nonpositive curvature, none deals with topology of open complete nonpositively curved manifolds, and this paper aims to fill the void. Most of the material discussed here is not widely-known. A number of questions is posed, ranging from naive to hopelessly difficult. Proofs are supplied when there is no explicit reference, and as always, the author is solely responsible for mistakes.

This survey has a narrow focus and does not discuss topological properties of

  1. • hyperbolic 3-manifolds [161, 112, 51],

  2. • open negatively pinched manifolds [19, 26],

  3. • non-Riemannian nonpositively curved manifolds [61],

  4. • compact nonpositively curved manifolds [78, 124],

  5. • higher rank locally symmetric spaces [125, 73, 28, 87] and their compactifications [35],

which are covered in the above references. We choose to work in the Riemannian setting which leads to some simplifications, even though many results hold in a far greater generality, and the references usually point to the strongest results available. Special attention is given to rank one manifolds, such as manifolds of negative curvature. Nonpositively curved manifolds are aspherical so we focus on groups of finite cohomological dimension (i.e. fundamental groups of aspherical manifolds), or better yet, groups of type F (i.e. the fundamental groups of compact aspherical manifolds with boundary).

Conventions: unless stated otherwise, manifolds are smooth, metrics are Riemannian, and sectional curvature is denoted by K.

Acknowledgments: The author is grateful for NSF support (DMS- 1105045). Thanks are due to Jim Davis, Denis Osin, Yunhui Wu, and the referee for correcting misstatements in the earlier version.

Flavors of negative curvature

A Hadamard manifold is a connected simply-connected complete manifold of K ≤ 0. By the Cartan-Hadamard theorem, any Hadamard manifold is diffeomorphic to a Euclidean space. Thus any complete manifold of K ≤ 0 is the quotient of a Hadamard manifold by a discrete torsion-free isometry group (torsion-freeness can be seen geometrically: any finite isometry group of a Hadamard manifold fixes the circumcenter of its orbit, or topologically: a nontrivial finite group has infinite cohomological dimension so it cannot act freely on a contractible manifold).

Type
Chapter
Information
Geometry, Topology, and Dynamics in Negative Curvature , pp. 32 - 83
Publisher: Cambridge University Press
Print publication year: 2016

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Abresch, U., and Schroeder, V. 1992. Graph manifolds, ends of negatively curvedspaces and the hyperbolic 120-cell space. J. Differential Geom., 35(2), 299–336.CrossRefGoogle Scholar
[2] Adams, S., and Ballmann, W. 1998. Amenable isometry groups of Hadamarspaces. Math. Ann., 312(1), 183–195.CrossRefGoogle Scholar
[3] Agol, I. Tameness of hyperbolic 3-manifolds. arXiv:math/0405568.
[4] Agol, I. 2013. The virtual Haken conjecture. Doc. Math., 18, 1045–1087. With an appendix by Agol, , D., Groves, and J., Manning.Google Scholar
[5] Alperin, R. C., and Shalen, P. B. 1982. Linear groups of finite cohomological dimension. Invent. Math., 66(1), 89–98.CrossRefGoogle Scholar
[6] Anderson, M. T. 1987. Metrics of negative curvature on vector bundles. Proc. Amer. Math. Soc., 99(2), 357–363.CrossRefGoogle Scholar
[7] Anderson, M. T. 2006. Dehn filling and Einstein metrics in higher dimensions. J. Differential Geom., 73(2), 219–261.CrossRefGoogle Scholar
[8] Ardanza-Trevijano Moras, S. I. 2000. Exotic smooth structures on negatively curved manifolds that are not of the homotopy type of a locally symmetric space. ProQuest LLC, Ann Arbor, MI. Thesis (Ph.D.)–SUNY at Binghamton.
[9] Arzhantseva, G., Bridson, M. R., Januszkiewicz, T., Leary, I. J., Minasyan, A., and Świa tkowski, J. 2009. Infinite groups with fixed point properties. Geom. Topol., 13(3), 1229–1263.CrossRefGoogle Scholar
[10] Aschenbrenner, M., Friedl, S., and Wilton, H. 3-manifold groups. arXiv:1205.0202.
[11] Ballmann, W. 1995. Lectures on spaces of nonpositive curvature. DMV Seminar, vol. 25. Basel: Birkhäuser Verlag. With an appendix by Misha Brin.CrossRefGoogle Scholar
[12] Ballmann, W., and Buyalo, S. 2008. Periodic rank one geodesics in Hadamard spaces. Pages 19–27 of: Geometric and probabilistic structures in dynamics. Contemp. Math., vol. 469. Providence, RI: Amer. Math. Soc.Google Scholar
[13] Ballmann, W., Gromov, M., , and Schroeder, V. 1985. Manifolds of nonpositive curvature. Progress in Mathematics, vol. 61. Boston, MA: Birkhäuser Boston Inc.CrossRefGoogle Scholar
[14] Balser, A., and Lytchak, A. 2005. Centers of convex subsets of buildings. Ann. Global Anal. Geom., 28(2), 201–209.
[15] Bartels, A., and Lück, W. 2012. The Borel conjecture for hyperbolic and CAT(0)- groups. Ann. of Math. (2), 175(2), 631–689.CrossRefGoogle Scholar
[16] Bartels, A., Farrell, F. T., and Lück, W. 2014. The Farrell-Jones Conjecture for cocompact lattices in virtually connected Lie groups. J. Amer. Math. Soc., 27(2), 339–388.CrossRefGoogle Scholar
[17] Bavard, C. 1991. Longueur stable des commutateurs. Enseign. Math. (2), 37(1-2), 109–150.Google Scholar
[18] Belegradek, I. Obstructions to nonpositive curvature for open manifolds. arXiv:1208.5220v1, to appear in Proc. Lond. Math. Soc.
[19] Belegradek, I. 2001. Pinching, Pontrjagin classes, and negatively curved vector bundles. Invent. Math., 144(2), 353–379.CrossRefGoogle Scholar
[20] Belegradek, I. 2007. Aspherical manifolds with relatively hyperbolic fundamental groups. Geom. Dedicata, 129, 119–144.CrossRefGoogle Scholar
[21] Belegradek, I. 2012a. Complex hyperbolic hyperplane complements. Math. Ann., 353(2), 545–579.CrossRefGoogle Scholar
[22] Belegradek, I. 2012b. Rigidity and relative hyperbolicity of real hyperbolic hyperplane complements. Pure Appl. Math. Q., 8(1), 15–51.CrossRefGoogle Scholar
[23] Belegradek, I. 2013. An assortment of negatively curved ends. J. Topol. Anal., 5(4), 439–449.CrossRefGoogle Scholar
[24] Belegradek, I., and Hruska, G. C. 2013. Hyperplane arrangements in negatively curved manifolds and relative hyperbolicity. Groups Geom. Dyn., 7(1), 13–38.CrossRefGoogle Scholar
[25] Belegradek, I., and Kapovitch, V. 2005. Pinching estimates for negatively curved manifolds with nilpotent fundamental groups. Geom. Funct. Anal., 15(5), 929– 938.CrossRefGoogle Scholar
[26] Belegradek, I., and Kapovitch, V. 2006. Classification of negatively pinched manifolds with amenable fundamental groups. Acta Math., 196(2), 229–260.CrossRefGoogle Scholar
[27] Belegradek, I., and Nguyen Phan, T. T. Non-aspherical ends and nonpositive curvature. arXiv:1212.3303.
[28] Bestvina, M., and Feighn, M. 2002. Proper actions of lattices on contractible manifolds. Invent. Math., 150(2), 237–256.CrossRefGoogle Scholar
[29] Bestvina, M., and Fujiwara, K. 2002. Bounded cohomology of subgroups of mapping class groups. Geom. Topol., 6, 69–89.CrossRefGoogle Scholar
[30] Bestvina, M., and Fujiwara, K. 2009. Acharacterization of higher rank symmetric spaces via bounded cohomology. Geom. Funct. Anal., 19(1), 11–40.CrossRefGoogle Scholar
[31] Bestvina, M., Kapovich, M., and Kleiner, B. 2002. Van Kampen's embedding obstruction for discrete groups. Invent. Math., 150(2), 219–235.CrossRefGoogle Scholar
[32] Bieri, R. 1981. Homological dimension of discrete groups. Second edn. Queen Mary College Mathematical Notes. London: Queen Mary College Department of Pure Mathematics.Google Scholar
[33] Bishop, R. L., and O&Neill, B. 1969. Manifolds of negative curvature. Trans. Amer. Math. Soc., 145, 1–49.CrossRefGoogle Scholar
[34] Bonahon, F., and Siebenmann, L. C. 1987. The characteristic toric splitting of irreducible compact 3-orbifolds. Math. Ann., 278(1-4), 441–479.CrossRefGoogle Scholar
[35] Borel, A., and Ji, L. 2005. Compactifications of symmetric and locally symmetric spaces. Pages 69–137 of: Lie theory. Progr. Math., vol. 229. Boston, MA: Birkhäuser Boston.Google Scholar
[36] Borel, A., and Serre, J.-P. 1973. Corners and arithmetic groups. Comment. Math. Helv., 48, 436–491. Avec un appendice: Arrondissement des variétés à coins, par A. Douady et L. Hérault.CrossRefGoogle Scholar
[37] Bowditch, B. H. 2010. Notes on tameness. Enseign. Math. (2), 56(3-4), 229–285.Google Scholar
[38] Bowditch, B. H. 2012. Relatively hyperbolic groups. Internat. J. Algebra Comput., 22(3), 1250016, 66.CrossRefGoogle Scholar
[39] Bridson, M. R. 2001. On the subgroups of semihyperbolic groups. Pages 85–111 of: Essays on geometry and related topics, Vol. 1, 2. Monogr. Enseign. Math., vol. 38. Geneva: Enseignement Math.Google Scholar
[40] Bridson, M. R., and Haefliger, A. 1999. Metric spaces of non-positive curvature. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319. Berlin: Springer-Verlag.CrossRefGoogle Scholar
[41] Brown, K. S. 1994. Cohomology of groups. Graduate Texts in Mathematics, vol. 87. New York: Springer-Verlag. Corrected reprint of the 1982 original.Google Scholar
[42] Bumagin, I. 2004. The conjugacy problem for relatively hyperbolic groups. Algebr. Geom. Topol., 4, 1013–1040.CrossRefGoogle Scholar
[43] Burger, M., and Monod, N. 2002. Continuous bounded cohomology and applications to rigidity theory. Geom. Funct. Anal., 12(2), 219–280.CrossRefGoogle Scholar
[44] Burger, M., and Schroeder, V. 1987. Amenable groups and stabilizers of measures on the boundary of a Hadamard manifold. Math. Ann., 276(3), 505–514.CrossRefGoogle Scholar
[45] Buyalo, S. V. 1993. An example of a four-dimensional manifold of negative curvature. Algebra i Analiz, 5(1), 193–199.Google Scholar
[46] Buyalo, S. V., and Kobel'skii, V. L. 1995. Geometrization of graph-manifolds. II. Isometric geometrization. Algebra i Analiz, 7(3), 96–117.Google Scholar
[47] Buyalo, S. V., and Kobel'skii, V. L. 1996. Cusp closing of hyperbolic manifolds. Geom. Dedicata, 59(2), 147–156.CrossRefGoogle Scholar
[48] Buyalo, S. V., and Svetlov, P. V. 2004. Topological and geometric properties of graph manifolds. Algebra i Analiz, 16(2), 3–68.Google Scholar
[49] Calegari, D., and Gabai, D. 2006. Shrinkwrapping and the taming of hyperbolic 3-manifolds. J. Amer. Math. Soc., 19(2), 385–446.CrossRefGoogle Scholar
[50] Canary, R. D. 2008. Marden's tameness conjecture: history and applications. Pages 137–162 of: Geometry, analysis and topology of discrete groups. Adv. Lect. Math. (ALM), vol. 6. Int. Press, Somerville, MA.Google Scholar
[51] Canary, R. D., and McCullough, D. 2004. Homotopy equivalences of 3-manifolds and deformation theory of Kleinian groups. Mem. Amer. Math. Soc., 172(812), xii+218.Google Scholar
[52] Caprace, P.-E., and Fujiwara, K. 2010. Rank-one isometries of buildings and quasi-morphisms of Kac-Moody groups. Geom. Funct. Anal., 19(5), 1296–1319.CrossRefGoogle Scholar
[53] Caprace, P.-E., and Lytchak, A. 2010. At infinity of finite-dimensional CAT(0) spaces. Math. Ann., 346(1), 1–21.CrossRefGoogle Scholar
[54] Caprace, P.-E., and Monod, N. 2009a. Isometry groups of non-positively curved spaces: discrete subgroups. J. Topol., 2(4), 701–746.Google Scholar
[55] Caprace, P.-E., and Monod, N. 2009b. Isometry groups of non-positively curved spaces: structure theory. J. Topol., 2(4), 661–700.Google Scholar
[56] Charney, R. M., and Davis, M. W. 1995. Strict hyperbolization. Topology, 34(2), 329–350.CrossRefGoogle Scholar
[57] Cheeger, J., and Gromov, M. 1985. Bounds on the von Neumann dimension of L2-cohomology and the Gauss-Bonnet theorem for open manifolds. J. Differential Geom., 21(1), 1–34.CrossRefGoogle Scholar
[58] Cheeger, J., and Gromov, M. 1991. Chopping Riemannian manifolds. Pages 85–94 of: Differential geometry. Pitman Monogr. Surveys Pure Appl. Math., vol. 52. Harlow: Longman Sci. Tech.Google Scholar
[59] Dafermos, M. 1997. Exhaustions of complete manifolds of bounded curvature. Pages 319–322 of: Tsing Hua lectures on geometry & analysis (Hsinchu, 1990–1991). Int. Press, Cambridge, MA.Google Scholar
[60] Dahmani, F., Guirardel, V., and Osin, D.Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces. http://arXiv:1111.7048v3.
[61] Davis, M.W. 2008. The geometry and topology of Coxeter groups. London Mathematical Society Monographs Series, vol. 32. Princeton, NJ: Princeton University Press.Google Scholar
[62] Davis, M. W., and Januszkiewicz, T. 1991. Hyperbolization of polyhedra. J. Differential Geom., 34(2), 347–388.CrossRefGoogle Scholar
[63] Davis, M.W., Januszkiewicz, T., and Weinberger, S. 2001. Relative hyperbolization and aspherical bordisms: an addendum to “Hyperbolization of polyhedra”. J. Differential Geom., 58(3), 535–541.CrossRefGoogle Scholar
[64] Deraux, M. 2005. A negatively curved Kähler threefold not covered by the ball. Invent. Math., 160(3), 501–525.CrossRefGoogle Scholar
[65] Despotovic, Z. 2006. Action dimension of mapping class groups. ProQuest LLC, Ann Arbor, MI. Thesis (Ph.D.)–The University of Utah.
[66] Dold, A. 1995. Lectures on algebraic topology. Classics in Mathematics. Berlin: Springer-Verlag. Reprint of the 1972 edition.CrossRefGoogle Scholar
[67] Drutu, C., and Sapir, M. 2005. Relatively hyperbolic groups with rapid decay property. Int. Math. Res. Not., 1181–1194.Google Scholar
[68] Drutu, C., and Sapir, M. V. 2008. Groups acting on tree-graded spaces and splittings of relatively hyperbolic groups. Adv. Math., 217(3), 1313–1367.CrossRefGoogle Scholar
[69] Duchesne, B.Superrigidity In Infinite Dimension And Finite Rank Via Harmonic Maps. arXiv:1206.1964v1.
[70] Eberlein, P. 1979. Surfaces of nonpositive curvature. Mem. Amer. Math. Soc., 20(218), x+90.Google Scholar
[71] Eberlein, P. 1980. Lattices in spaces of nonpositive curvature. Ann. of Math. (2), 111(3), 435–476.CrossRefGoogle Scholar
[72] Eberlein, P., and O'Neill, B. 1973. Visibility manifolds. Pacific J. Math., 46, 45–109.CrossRefGoogle Scholar
[73] Eberlein, P. B. 1996. Geometry of nonpositively curved manifolds. Chicago Lectures in Mathematics. Chicago, IL: University of Chicago Press.Google Scholar
[74] Eymard, P. 1972. Moyennes invariantes et représentations unitaires. Lecture Notes in Mathematics, Vol. 300. Berlin: Springer-Verlag.CrossRefGoogle Scholar
[75] Farb, B. 1998. Relatively hyperbolic groups. Geom. Funct. Anal., 8(5), 810–840.CrossRefGoogle Scholar
[76] Farrell, F. T., and Jones, L. E. 1986. K-theory and dynamics. I. Ann. of Math. (2), 124(3), 531–569.Google Scholar
[77] Farrell, F. T., and Ontaneda, P. 2006. Branched covers of hyperbolic manifolds and harmonic maps. Comm. Anal. Geom., 14(2), 249–268.CrossRefGoogle Scholar
[78] Farrell, F. T., Jones, L. E., and Ontaneda, P. 2007. Negative curvature and exotic topology. Pages 329–347 of: Surveys in differential geometry. Vol. XI. Surv. Differ. Geom., vol. 11. Int. Press, Somerville, MA.Google Scholar
[79] Fornari, S., and Schroeder, V. 1990. Ramified coverings with nonpositive curvature. Math. Z., 203(1), 123–128.CrossRefGoogle Scholar
[80] Freedman, M. H., and Gabai, D. 2007. Covering a nontaming knot by the unlink. Algebr. Geom. Topol., 7, 1561–1578.CrossRefGoogle Scholar
[81] Fujiwara, K. 1988. A construction of negatively curved manifolds. Proc. Japan Acad. Ser. A Math. Sci., 64(9), 352–355.CrossRefGoogle Scholar
[82] Fujiwara, K., Nagano, K., and Shioya, T. 2006. Fixed point sets of parabolic isometries of CAT(0)-spaces. Comment. Math. Helv., 81(2), 305–335.Google Scholar
[83] Fukaya, K. 1984. A finiteness theorem for negatively curved manifolds. J. Differential Geom., 20(2), 497–521.CrossRefGoogle Scholar
[84] Fukaya, K., and Yamaguchi, T. 1992. The fundamental groups of almost nonnegatively curved manifolds. Ann. of Math. (2), 136(2), 253–333.CrossRefGoogle Scholar
[85] Furman, A. 2011. A survey of measured group theory. Pages 296–374 of: Geometry, rigidity, and group actions. Chicago Lectures in Math. Chicago, IL: Univ. Chicago Press.Google Scholar
[86] Gelander, T. 2004. Homotopy type and volume of locally symmetric manifolds. Duke Math. J., 124(3), 459–515.CrossRefGoogle Scholar
[87] Gelander, T. 2011. Volume versus rank of lattices. J. Reine Angew. Math., 661, 237–248.Google Scholar
[88] Goldman, M. E. 1971. An algebraic classification of noncompact 2-manifolds. Trans. Amer. Math. Soc., 156, 241–258.Google Scholar
[89] Gromoll, D., and Wolf, J. A. 1971. Some relations between the metric structure and the algebraic structure of the fundamental group in manifolds of nonpositive curvature. Bull. Amer. Math. Soc., 77, 545–552.CrossRefGoogle Scholar
[90] Gromov, M. 1978. Manifolds of negative curvature. J. Differential Geom., 13(2), 223–230.CrossRefGoogle Scholar
[91] Gromov, M. 1982. Volume and bounded cohomology. Inst. Hautes Études Sci. Publ. Math., 5–99 (1983).Google Scholar
[92] Gromov, M. 1987. Hyperbolic groups. Pages 75–263 of: Essays in group theory. Math. Sci. Res. Inst. Publ., vol. 8. New York: Springer.Google Scholar
[93] Gromov, M. 1993. Asymptotic invariants of infinite groups. Pages 1–295 of: Geometric group theory, Vol. 2 (Sussex, 1991). London Math. Soc. Lecture Note Ser., vol. 182. Cambridge: Cambridge Univ. Press.Google Scholar
[94] Gromov, M. 2003. Random walk in random groups. Geom. Funct. Anal., 13(1), 73–146.CrossRefGoogle Scholar
[95] Gromov, M., and Thurston, W. 1987. Pinching constants for hyperbolic manifolds. Invent. Math., 89(1), 1–12.CrossRefGoogle Scholar
[96] Groves, D., and Manning, J. F. 2008. Dehn filling in relatively hyperbolic groups. Israel J. Math., 168, 317–429.CrossRefGoogle Scholar
[97] Guilbault, C. R. 2007. Products of open manifolds with R. Fund. Math., 197, 197–214.CrossRefGoogle Scholar
[98] Hamenstädt, U. 2008. Bounded cohomology and isometry groups of hyperbolic spaces. J. Eur. Math. Soc. (JEMS), 10(2), 315–349.Google Scholar
[99] Hamenstädt, U. 2009a. Isometry groups of proper hyperbolic spaces. Geom. Funct. Anal., 19(1), 170–205.CrossRefGoogle Scholar
[100] Hamenstädt, U. 2009b. Rank-one isometries of proper CAT(0)-spaces. Pages 43–59 of: Discrete groups and geometric structures. Contemp. Math., vol. 501. Providence, RI: Amer. Math. Soc.Google Scholar
[101] Hamenstädt, U. 2012. Isometry groups of proper CAT(0)-spaces of rank one. Groups Geom. Dyn., 6(3), 579–618.CrossRefGoogle Scholar
[102] Hatcher, A.Basic 3-Manifold Topology. www.math.cornell.edu/ hatcher/.
[103] Hattori, T. 1996. Asymptotic geometry of arithmetic quotients of symmetric spaces. Math. Z., 222(2), 247–277.CrossRefGoogle Scholar
[104] Hersonsky, S., and Paulin, F. 1996. On the volumes of complex hyperbolic manifolds. Duke Math. J., 84(3), 719–737.CrossRefGoogle Scholar
[105] Hull, M., and Osin, D. 2013. Induced quasicocycles on groups with hyperbolically embedded subgroups. Algebr. Geom. Topol., 13(5), 2635–2665.CrossRefGoogle Scholar
[106] Hummel, C., and Schroeder, V. 1996. Cusp closing in rank one symmetric spaces. Invent. Math., 123(2), 283–307.CrossRefGoogle Scholar
[107] Izeki, H., and Nayatani, S. 2005. Combinatorial harmonic maps and discretegroup actions on Hadamard spaces. Geom. Dedicata, 114, 147–188.CrossRefGoogle Scholar
[108] Izeki, H., and Nayatani, S. 2010. An approach to superrigidity and fixed-point theorems via harmonic maps. Pages 135–160 of: Selected papers on analysis and differential equations. Amer. Math. Soc. Transl. Ser. 2, vol. 230. Providence, RI: Amer. Math. Soc.
[109] Izeki, H., Kondo, T., and Nayatani, S. 2012. N-step energy of maps and the fixed-point property of random groups. Groups Geom. Dyn., 6(4), 701–736.CrossRefGoogle Scholar
[110] Ji, L., and MacPherson, R. 2002. Geometry of compactifications of locally symmetric spaces. Ann. Inst. Fourier (Grenoble), 52(2), 457–559.CrossRefGoogle Scholar
[111] Johnson, B. E. 1972. Cohomology in Banach algebras. Providence, R.I.: American Mathematical Society. Memoirs of the American Mathematical Society, No. 127.Google Scholar
[112] Kapovich, M. 2001. Hyperbolic manifolds and discrete groups. Progress in Mathematics, vol. 183. Boston, MA: Birkhäuser Boston Inc.Google Scholar
[113] Kapovich, M., and Leeb, B. 1996. Actions of discrete groups on nonpositively curved spaces. Math. Ann., 306(2), 341–352.CrossRefGoogle Scholar
[114] Kapovitch, V., and Wilking, B.Structure of fundamental groups of manifolds with Ricci curvature bounded below. arXiv:1105.5955.
[115] Kapovitch, V., Petrunin, A., and Tuschmann, W. 2010. Nilpotency, almost nonnegative curvature, and the gradient flow on Alexandrov spaces. Ann. of Math. (2), 171(1), 343–373.CrossRefGoogle Scholar
[116] Karlsson, A., and Noskov, G. A. 2004. Some groups having only elementary actions on metric spaces with hyperbolic boundaries. Geom. Dedicata, 104, 119–137.CrossRefGoogle Scholar
[117] Kerékjártó, B. 1923. Vorlesungen über Topologie, I. Berlin: Springer.CrossRefGoogle Scholar
[118] Kleiner, B. 1999. Integrating infinitesimal flats in Hadamard manifolds. preprint.
[119] Lawson, Jr., H. B., and Yau, S. T. 1972. Compact manifolds of nonpositive curvature. J. Differential Geometry, 7, 211–228.CrossRefGoogle Scholar
[120] Leeb, B. 1995. 3-manifolds with(out) metrics of nonpositive curvature. Invent. Math., 122(2), 277–289.CrossRefGoogle Scholar
[121] Leuzinger, E. 1995. An exhaustion of locally symmetric spaces by compact submanifolds with corners. Invent. Math., 121(2), 389–410.CrossRefGoogle Scholar
[122] Leuzinger, E. 2004. On polyhedral retracts and compactifications of locally symmetric spaces. Differential Geom. Appl., 20(3), 293–318.CrossRefGoogle Scholar
[123] Liu, Yi. 2013. Virtual cubulation of nonpositively curved graph manifolds. J. Topol., 6(4), 793–822.CrossRefGoogle Scholar
[124] Lück, W. 2010. Survey on aspherical manifolds. Pages 53–82 of: European Congress of Mathematics. Eur. Math. Soc., Zürich.Google Scholar
[125] Margulis, G. A. 1991. Discrete subgroups of semisimple Lie groups. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 17. Berlin: Springer-Verlag.CrossRefGoogle Scholar
[126] Minasyan, A., and Osin, D.Acylindrically hyperbolic groups acting on trees. in preparation.
[127] Mineyev, I., and Yaman, A.Relative hyperbolicity and bounded cohomology. www.math.uiuc.edu/~mineyev/math/.
[128] Mineyev, I., Monod, N., and Shalom, Y. 2004. Ideal bicombings for hyperbolic groups and applications. Topology, 43(6), 1319–1344.CrossRefGoogle Scholar
[129] Monod, N. 2006a. An invitation to bounded cohomology. Pages 1183–1211 of: International Congress of Mathematicians. Vol. II. Eur. Math. Soc., Zürich.Google Scholar
[130] Monod, N. 2006b. Superrigidity for irreducible lattices and geometric splitting. J. Amer. Math. Soc., 19(4), 781–814.CrossRefGoogle Scholar
[131] Monod, N., and Popa, S. 2003. On co-amenability for groups and von Neumann algebras. C. R. Math. Acad. Sci. Soc. R. Can., 25(3), 82–87.Google Scholar
[132] Monod, N., and Shalom, Y. 2004. Cocycle superrigidity and bounded cohomology for negatively curved spaces. J. Differential Geom., 67(3), 395–455.CrossRefGoogle Scholar
[133] Monod, N., and Shalom, Y. 2006. Orbit equivalence rigidity and bounded cohomology. Ann. of Math. (2), 164(3), 825–878.CrossRefGoogle Scholar
[134] Moon, S. 2009. Amenable actions of discrete groups. Ph.D. thesis, Universite de Neuchatel, http://doc.rero.ch/record/18088.
[135] Morris, D. Introduction to arithmetic groups. arXiv:math/0106063.
[136] Mostow, G. D., and Siu, Y. T. 1980. A compact Kähler surface of negative curvature not covered by the ball. Ann. of Math. (2), 112(2), 321–360.CrossRefGoogle Scholar
[137] Naor, A., and Silberman, L. 2011. Poincarée inequalities, embeddings, and wild groups. Compos. Math., 147(5), 1546–1572.CrossRefGoogle Scholar
[138] Nguyẽn Phan, T. T. Nil happens. What about Sol? arXiv:1207.1734.
[139] Nguyẽn Phan, T. T. On finite volume, negatively curved manifolds. arXiv:1110.4087.
[140] Nguyẽn Phan, T. T. 2013. Nonpositively curved manifolds containing a prescribed nonpositively curved hypersurface. Topology Proc., 42, 39–41.Google Scholar
[141] Noskov, G. A. 1990. Bounded cohomology of discrete groups with coefficients. Algebra i Analiz, 2(5), 146–164.Google Scholar
[142] Ontaneda, P. Pinched smooth hyperbolization. arXiv:1110.6374v1.
[143] Ontaneda, P. 2003. The double of a hyperbolic manifold and non-positively curved exotic PL structures. Trans. Amer. Math. Soc., 355(3), 935–965 (electronic).CrossRefGoogle Scholar
[144] Osin, D. V. Acylindrically hyperbolic groups. arXiv:1304.1246v1.
[145] Osin, D. V. 2005. Asymptotic dimension of relatively hyperbolic groups. Int. Math. Res. Not., 2143–2161.Google Scholar
[146] Osin, D. V. 2006. Relatively hyperbolic groups: intrinsic geometry, algebraic properties, and algorithmic problems. Mem. Amer. Math. Soc., 179(843), vi+100.Google Scholar
[147] Osin, D.V. 2007. Peripheral fillings of relatively hyperbolic groups. Invent. Math., 167(2), 295–326.CrossRefGoogle Scholar
[148] Ould Houcine, A. 2007. Embeddings in finitely presented groups which preserve the center. J. Algebra, 307(1), 1–23.CrossRefGoogle Scholar
[149] Rebbechi, D. Y. 2001. Algorithmic properties of relatively hyperbolic groups. Ph.D. thesis, Rutgers Newark arXiv:math/0302245v1.
[150] Richards, I. 1963. On the classification of noncompact surfaces. Trans. Amer. Math. Soc., 106, 259–269.CrossRefGoogle Scholar
[151] Saper, L. 1997. Tilings and finite energy retractions of locally symmetric spaces. Comment. Math. Helv., 72(2), 167–202.CrossRefGoogle Scholar
[152] Schoen, R., and Yau, S.-T. 1994. Lectures on differential geometry. Conference Proceedings and Lecture Notes in Geometry and Topology, I. Cambridge, MA: International Press.Google Scholar
[153] Schroeder, V. 1985. A splitting theorem for spaces of nonpositive curvature. Invent. Math., 79(2), 323–327.CrossRefGoogle Scholar
[154] Schroeder, V. 1989. A cusp closing theorem. Proc. Amer. Math. Soc., 106(3), 797–802.CrossRefGoogle Scholar
[155] Schroeder, V. 1991. Analytic manifolds of nonpositive curvature with higher rank subspaces. Arch. Math. (Basel), 56(1), 81–85.CrossRefGoogle Scholar
[156] Scott, P., and Tucker, T. 1989. Some examples of exotic noncompact 3-manifolds. Quart. J. Math. Oxford Ser. (2), 40(160), 481–499.CrossRefGoogle Scholar
[157] Siebenmann, L. C. Classifcation des 2-variéetés ouvertes. preprint, 2008.
[158] Siebenmann, L. C. 1969. On detecting open collars. Trans. Amer. Math. Soc., 142, 201–227.CrossRefGoogle Scholar
[159] Siebenmann, L. C. 1970. A total Whitehead torsion obstruction to fibering over the circle. Comment. Math. Helv., 45, 1–48.CrossRefGoogle Scholar
[161] Thurston, W. P.The geometry and topology of three-manifolds. Princeton lecture notes (1978-1981).
[162] Tukia, P. 1994. Convergence groups and Gromov's metric hyperbolic spaces. New Zealand J. Math., 23(2), 157–187.Google Scholar
[163] Wang, M.-T., and Lin, C.-H. 1997. A note on the exhaustion function for complete manifolds. Pages 269–277 of: Tsing Hua lectures on geometry & analysis (Hsinchu, 1990–1991). Int. Press, Cambridge, MA.Google Scholar
[164] Wu, Y. Translation lengths of parabolic isometries of CAT(0) spaces and their applications. http://math.rice.edu/ yw22/.
[165] Yoon, S. Y. 2004. A lower bound to the action dimension of a group. Algebr. Geom. Topol., 4, 273–296 (electronic).CrossRefGoogle Scholar
[166] Zheng, F. 1996. Examples of non-positively curved Kähler manifolds. Comm. Anal. Geom., 4(1-2), 129–160.CrossRefGoogle Scholar

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×