The Borel Conjecture for hyperbolic and CAT(0)-groups

Abstract

We prove the Borel Conjecture for a class of groups containing word-hyperbolic groups and groups acting properly, isometrically and cocompactly on a finite-dimensional CAT(0)-space.

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    @incollection {Farrell-Jones(1993c), MRKEY = {1216623},
      AUTHOR = {Farrell, F. T. and Jones, L. E.},
      TITLE = {Topological rigidity for compact non-positively curved manifolds},
      BOOKTITLE = {Differential Geometry: {R}iemannian Geometry},
      VENUE={{L}os {A}ngeles, {CA},
      1990},
      SERIES = {Proc. Sympos. Pure Math.},
      VOLUME = {54},
      PAGES = {229--274},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {1993},
      MRCLASS = {57R65 (53C21 57R67 57S30)},
      MRNUMBER = {1216623},
      MRREVIEWER = {Christopher W. Stark},
      ZBLNUMBER = {0796.53043},
     }
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    @article {Farrell-Jones(1998), MRKEY = {1639544},
      AUTHOR = {Farrell, F. T. and Jones, L. E.},
      TITLE = {Rigidity for aspherical manifolds with {$\pi\sb 1\subset{\it GL}\sb m(\mathbb{R})$}},
      JOURNAL = {Asian J. Math.},
      FJOURNAL = {The Asian Journal of Mathematics},
      VOLUME = {2},
      YEAR = {1998},
      NUMBER = {2},
      PAGES = {215--262},
      ISSN = {1093-6106},
      MRCLASS = {57R57 (53C20)},
      MRNUMBER = {1639544},
      MRREVIEWER = {A. A. Ranicki},
      ZBLNUMBER = {0912.57012},
      }
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      AUTHOR = {Freedman, Michael H.},
      TITLE = {The disk theorem for four-dimensional manifolds},
      BOOKTITLE = {Proceedings of the {I}nternational {C}ongress of {M}athematicians, {V}ol. 1, 2},
      VENUE={{W}arsaw, 1983},
      PAGES = {647--663},
      PUBLISHER = {PWN},
      ADDRESS = {Warsaw},
      YEAR = {1984},
      MRCLASS = {57N15 (57R65)},
      MRNUMBER = {0804721},
      MRREVIEWER = {Frank Quinn},
      ZBLNUMBER = {0577.57003},
      }
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    @article {Gersten(1967), MRKEY = {0212813},
      AUTHOR = {Gersten, Stephen M.},
      TITLE = {The torsion of a self equivalence},
      JOURNAL = {Topology},
      FJOURNAL = {Topology. An International Journal of Mathematics},
      VOLUME = {6},
      YEAR = {1967},
      PAGES = {411--414},
      ISSN = {0040-9383},
      MRCLASS = {57.10},
      MRNUMBER = {0212813},
      MRREVIEWER = {J.-P. Meyer},
      ZBLNUMBER = {0153.25003},
      DOI = {10.1016/0040-9383(67)90027-4},
     }
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      AUTHOR = {Gromov, M.},
      TITLE = {Hyperbolic groups},
      BOOKTITLE = {Essays in Group Theory},
      SERIES = {Math. Sci. Res. Inst. Publ.},
      VOLUME = {8},
      PAGES = {75--263},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1987},
      MRCLASS = {20F32 (20F06 20F10 22E40 53C20 57R75 58F17)},
      MRNUMBER = {0919829},
      MRREVIEWER = {Christopher W. Stark},
      ZBLNUMBER = {0634.20015},
      }
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    @article {Higson-Lafforgue-Skandalis(2002), MRKEY = {1911663},
      AUTHOR = {Higson, N. and Lafforgue, V. and Skandalis, G.},
      TITLE = {Counterexamples to the {B}aum-{C}onnes conjecture},
      JOURNAL = {Geom. Funct. Anal.},
      FJOURNAL = {Geometric and Functional Analysis},
      VOLUME = {12},
      YEAR = {2002},
      NUMBER = {2},
      PAGES = {330--354},
      ISSN = {1016-443X},
      CODEN = {GFANFB},
      MRCLASS = {19K56 (22A22 46L80 46L85 58J22)},
      MRNUMBER = {1911663},
      MRREVIEWER = {Alain Valette},
      DOI = {10.1007/s00039-002-8249-5},
      ZBLNUMBER = {1014.46043},
      }
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    @article {Hu(1993), MRKEY = {1243784},
      AUTHOR = {Hu, Bi Zhong},
      TITLE = {Whitehead groups of finite polyhedra with nonpositive curvature},
      JOURNAL = {J. Differential Geom.},
      FJOURNAL = {Journal of Differential Geometry},
      VOLUME = {38},
      YEAR = {1993},
      NUMBER = {3},
      PAGES = {501--517},
      ISSN = {0022-040X},
      CODEN = {JDGEAS},
      MRCLASS = {57Q10 (19J10 53C21 57R50)},
      MRNUMBER = {1243784},
      MRREVIEWER = {Tadeusz Januszkiewicz},
      URL = {http://projecteuclid.org/getRecord?id=euclid.jdg/1214454480},
      ZBLNUMBER = {0781.57009},
      }
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    @article {Ji(2007SarithI), MRKEY = {2353862},
      AUTHOR = {Ji, Lizhen},
      TITLE = {The integral {N}ovikov conjectures for {$S$}-arithmetic groups. {I}},
      JOURNAL = {$K$-Theory},
      FJOURNAL = {$K$-Theory. An Interdisciplinary Journal for the Development, Application, and Influence of $K$-Theory in the Mathematical Sciences},
      VOLUME = {38},
      YEAR = {2007},
      NUMBER = {1},
      PAGES = {35--47},
      ISSN = {0920-3036},
      CODEN = {KTHEEO},
      MRCLASS = {57R19 (20G30)},
      MRNUMBER = {2353862},
      MRREVIEWER = {Boris Goldfarb},
      DOI = {10.1007/s10977-007-9007-0},
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      }
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    @article {Kreck-Lueck(2009nonasph), MRKEY = {2532709},
      AUTHOR = {Kreck, M. and L{ü}ck, W.},
      TITLE = {Topological rigidity for non-aspherical manifolds},
      JOURNAL = {Pure Appl. Math. Q.},
      FJOURNAL = {Pure and Applied Mathematics Quarterly},
      VOLUME = {5},
      YEAR = {2009},
      NUMBER = {3, Special Issue: In honor of Friedrich Hirzebruch. Part 2},
      PAGES = {873--914},
      ISSN = {1558-8599},
      MRCLASS = {57N99 (57R67)},
      MRNUMBER = {2532709},
      MRREVIEWER = {A. A. Ranicki},
      ZBLNUMBER = {1196.57018},
     }
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    @article {Lueck(1986), MRKEY = {0873491},
      AUTHOR = {L{ü}ck, Wolfgang},
      TITLE = {The transfer maps induced in the algebraic {$K\sb 0$}-and {$K\sb 1$}-groups by a fibration. {I}},
      JOURNAL = {Math. Scand.},
      FJOURNAL = {Mathematica Scandinavica},
      VOLUME = {59},
      YEAR = {1986},
      NUMBER = {1},
      PAGES = {93--121},
      ISSN = {0025-5521},
      CODEN = {MTSCAN},
      MRCLASS = {18F25 (19M05 55R12)},
      MRNUMBER = {0873491},
      MRREVIEWER = {Gunnar Carlsson},
      ZBLNUMBER = {0589.57020},
     }
  • [Lueck(1987)] Go to document W. Lück, "The transfer maps induced in the algebraic $K_0$- and $K_1$-groups by a fibration. II," J. Pure Appl. Algebra, vol. 45, iss. 2, pp. 143-169, 1987.
    @article {Lueck(1987), MRKEY = {0889589},
      AUTHOR = {L{ü}ck, Wolfgang},
      TITLE = {The transfer maps induced in the algebraic {$K\sb 0$}- and {$K\sb 1$}-groups by a fibration. {II}},
      JOURNAL = {J. Pure Appl. Algebra},
      FJOURNAL = {Journal of Pure and Applied Algebra},
      VOLUME = {45},
      YEAR = {1987},
      NUMBER = {2},
      PAGES = {143--169},
      ISSN = {0022-4049},
      CODEN = {JPAAA2},
      MRCLASS = {18F25 (19A31 19B28 55R12)},
      MRNUMBER = {0889589},
      MRREVIEWER = {Gunnar Carlsson},
      DOI = {10.1016/0022-4049(87)90066-1},
      ZBLNUMBER = {0657.57010},
     }
  • [Lueck(1989)] Go to document W. Lück, Transformation Groups and Algebraic $K$-theory, New York: Springer-Verlag, 1989, vol. 1408.
    @book {Lueck(1989), MRKEY = {1027600},
      AUTHOR = {L{ü}ck, Wolfgang},
      TITLE = {Transformation Groups and Algebraic {$K$}-theory},
      SERIES = {Lecture Notes in Math.},
      VOLUME = {1408},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1989},
      PAGES = {xii+443},
      ISBN = {3-540-51846-0},
      MRCLASS = {57Sxx (18F25 19-02 57Q10 57Q12)},
      MRNUMBER = {1027600},
      MRREVIEWER = {A. A. Ranicki},
      ZBLNUMBER = {0679.57022},
      DOI = {10.1007/BFb0083681},
     }
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    @article {Lueck(2005heis), MRKEY = {2175154},
      AUTHOR = {L{ü}ck, Wolfgang},
      TITLE = {{$K$}- and {$L$}-theory of the semi-direct product of the discrete 3-dimensional {H}eisenberg group by {${\Bbb Z}/4$}},
      JOURNAL = {Geom. Topol.},
      FJOURNAL = {Geometry and Topology},
      VOLUME = {9},
      YEAR = {2005},
      PAGES = {1639--1676},
      ISSN = {1465-3060},
      MRCLASS = {19K35 (19A31 19B28 19D35 19G24)},
      MRNUMBER = {2175154},
      MRREVIEWER = {Erik K. Pedersen},
      DOI = {10.2140/gt.2005.9.1639},
      ZBLNUMBER = {1073.19004},
     }
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      AUTHOR = {L{ü}ck, Wolfgang and Ranicki, A.},
      TITLE = {Surgery transfer},
      BOOKTITLE = {Algebraic Topology and Transformation Groups},
      VENUE={{G}öttingen, 1987},
      SERIES = {Lecture Notes in Math.},
      VOLUME = {1361},
      PAGES = {167--246},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1988},
      MRCLASS = {57R67 (19D99 55R12 57R91)},
      MRNUMBER = {0979509},
      MRREVIEWER = {Ian Hambleton},
      DOI = {10.1007/BFb0083036},
      ZBLNUMBER = {0677.57012},
     }
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    @incollection{Lueck-Reich(2005),
      author={L{ü}ck, Wolfgang and Reich, H.},
      TITLE = {The {B}aum-{C}onnes and the {F}arrell-{J}ones conjectures in {$K$}- and {$L$}-theory},
      BOOKTITLE={Handbook of $K$-Theory},
      VOLUME={1, 2},
      PAGES={703--842},
      PUBLISHER={Springer-Verlag},
      ADDRESS={New York},
      YEAR={2005},
      MRNUMBER = {2181833},
      ZBLNUMBER = {1120.19001},
      DOI = {10.1007/978-3-540-27855-9_15},
     }
  • [Mineyev(2005)] Go to document I. Mineyev, "Flows and joins of metric spaces," Geom. Topol., vol. 9, pp. 403-482, 2005.
    @article {Mineyev(2005), MRKEY = {2140987},
      AUTHOR = {Mineyev, Igor},
      TITLE = {Flows and joins of metric spaces},
      JOURNAL = {Geom. Topol.},
      FJOURNAL = {Geometry and Topology},
      VOLUME = {9},
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      PAGES = {403--482},
      ISSN = {1465-3060},
      MRCLASS = {37D40 (20F65 20F67 57M07 57N16)},
      MRNUMBER = {2140987},
      MRREVIEWER = {Darryl McCullough},
      DOI = {10.2140/gt.2005.9.403},
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      }
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    @article {Mineyev-Yu(2002), MRKEY = {1914618},
      AUTHOR = {Mineyev, Igor and Yu, Guoliang},
      TITLE = {The {B}aum-{C}onnes conjecture for hyperbolic groups},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {149},
      YEAR = {2002},
      NUMBER = {1},
      PAGES = {97--122},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {20F67 (19K56 20F65 22D25 46L89)},
      MRNUMBER = {1914618},
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      ZBLNUMBER = {1038.20030},
      }
  • [Moussong(1987)] Go to document G. Moussong, Hyperbolic Coxeter groups, 1987.
    @misc{Moussong(1987),
      author={Moussong, G.},
      TITLE={Hyperbolic {C}oxeter groups},
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      YEAR={1987},
      URL = {www.math.osu.edu/~mdavis/papers/Moussongthesis.pdf},
     }
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    @article {Olshanskii(1979), MRKEY = {0567039},
      AUTHOR = {Ol{$'$}{š}anski{\u\i},
      A. Ju.},
      TITLE = {An infinite simple torsion-free {N}oetherian group},
      JOURNAL = {Izv. Akad. Nauk SSSR Ser. Mat.},
      FJOURNAL = {Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya},
      VOLUME = {43},
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      ISSN = {0373-2436},
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     }
  • [Olshanskii-Osin-Sapir(2009)] Go to document A. Y. Ol$’$shanskii, D. V. Osin, and M. V. Sapir, "Lacunary hyperbolic groups," Geom. Topol., vol. 13, iss. 4, pp. 2051-2140, 2009.
    @article {Olshanskii-Osin-Sapir(2009), MRKEY = {2507115},
      AUTHOR = {Ol{$'$}shanskii, Alexander Yu. and Osin, Denis V. and Sapir, Mark V.},
      TITLE = {Lacunary hyperbolic groups},
      NOTE = {with an appendix by Michael Kapovich and Bruce Kleiner},
      JOURNAL = {Geom. Topol.},
      FJOURNAL = {Geometry \& Topology},
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      YEAR = {2009},
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      PAGES = {2051--2140},
      ISSN = {1465-3060},
      MRCLASS = {20F67 (20F65 20F69)},
      MRNUMBER = {2507115},
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     }
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    @book {Paulin(2004), MRKEY = {2111650},
      AUTHOR = {Paulin, Fr{é}d{é}ric},
      TITLE = {Sur la Théorie Élémentaire des Groupes Libres (d'après {S}ela)},
      SERIES = {Astérisque},
      FJOURNAL = {Astérisque},
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      YEAR = {2004},
      PAGES = {ix, 363--402},
      ISSN = {0303-1179},
      MRCLASS = {20A15 (03B25 20E05 20E08 20F65)},
      MRNUMBER = {2111650},
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      AUTHOR = {Pedersen, Erik K. and Weibel, Charles A.},
      TITLE = {A nonconnective delooping of algebraic {$K$}-theory},
      BOOKTITLE = {Algebraic and Geometric Topology},
      VENUE={{N}ew {B}runswick, {N}.{J}., 1983},
      SERIES = {Lecture Notes in Math.},
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      YEAR = {1985},
      MRCLASS = {18F25 (19D06 19D35)},
      MRNUMBER = {0802790},
      MRREVIEWER = {Ross Staffeldt},
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      }
  • [Quillen(1973)] Go to document D. Quillen, "Higher algebraic $K$-theory: I," in Higher $K$-Theories, New York: Springer-Verlag, 1973, vol. 341, pp. 85-147.
    @incollection {Quillen(1973), MRKEY = {0338129},
      AUTHOR = {Quillen, Daniel},
      TITLE = {Higher algebraic {$K$}-theory: {I}},
      BOOKTITLE = {Higher $K$-Theories},
      VENUE={{P}roc. {C}onf., {B}attelle {M}emorial {I}nst., {S}eattle, {W}ash., 1972},
      PAGES = {85--147},
      SERIES={Lecture Notes in Math.},
      VOLUME={341},
      PUBLISHER = {Springer-Verlag},
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      MRCLASS = {18F25},
      MRNUMBER = {0338129},
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     }
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    @incollection {Quinn(2005), MRKEY = {0780593},
      AUTHOR = {Quinn, Frank},
      TITLE = {The embedding theorem for towers},
      BOOKTITLE = {Four-Manifold Theory},
      VENUE={{D}urham, {N}.{H}., 1982)},
      SERIES = {Contemp. Math.},
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      PAGES = {461--471},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {1984},
      MRCLASS = {57N13},
      MRNUMBER = {0780593},
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      }
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    @book {Ranicki(1981), MRKEY = {0620795},
      AUTHOR = {Ranicki, Andrew},
      TITLE = {Exact Sequences in the Algebraic Theory of Surgery},
      SERIES = {Math. Notes},
      VOLUME = {26},
      PUBLISHER = {Princeton Univ. Press},
      ADDRESS = {Princeton, N.J.},
      YEAR = {1981},
      PAGES = {xvii+864},
      ISBN = {0-691-08276-6},
      MRCLASS = {57R67 (18F25 57-02)},
      MRNUMBER = {0620795},
      MRREVIEWER = {F. T. Farrell},
      ZBLNUMBER = {0471.57012},
      }
  • [Ranicki(1985)] A. Ranicki, "The algebraic theory of finiteness obstruction," Math. Scand., vol. 57, iss. 1, pp. 105-126, 1985.
    @article {Ranicki(1985), MRKEY = {0815431},
      AUTHOR = {Ranicki, Andrew},
      TITLE = {The algebraic theory of finiteness obstruction},
      JOURNAL = {Math. Scand.},
      FJOURNAL = {Mathematica Scandinavica},
      VOLUME = {57},
      YEAR = {1985},
      NUMBER = {1},
      PAGES = {105--126},
      ISSN = {0025-5521},
      CODEN = {MTSCAN},
      MRCLASS = {18F25 (19A31 57Q12)},
      MRNUMBER = {0815431},
      MRREVIEWER = {Ian Hambleton},
      ZBLNUMBER = {0589.57018},
      }
  • [Ranicki(1985_torsI)] Go to document A. Ranicki, "The algebraic theory of torsion. I. Foundations," in Algebraic and Geometric Topology, New York: Springer-Verlag, 1985, vol. 1126, pp. 199-237.
    @incollection {Ranicki(1985_torsI), MRKEY = {0802792},
      AUTHOR = {Ranicki, Andrew},
      TITLE = {The algebraic theory of torsion. {I}. {F}oundations},
      BOOKTITLE = {Algebraic and Geometric Topology},
      VENUE={{N}ew {B}runswick, {N}.{J}., 1983},
      SERIES = {Lecture Notes in Math.},
      VOLUME = {1126},
      PAGES = {199--237},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1985},
      MRCLASS = {18F25 (19B28 57Q10 57R67)},
      MRNUMBER = {0802792},
      MRREVIEWER = {Julius L. Shaneson},
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Authors

Arthur Bartels

Mathematisches Institut
Westfälische Wilhelms-Universität Münster
48149 Münster
Germany

Wolfgang Lück

HIM (Hausdorff Research Institute for Mathematics)
Poppelsdorfer Allee 45
53115 Bonn
Germany

and

Rheinische Friedrich-Wilhelms-Universität Bonn
Mathematisches Institut
Endenicher Allee 60
53115 Bonn
Germany