Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)

 
 

 

The Farrell-Jones Conjecture for cocompact lattices in virtually connected Lie groups


Authors: A. Bartels, F. T. Farrell and W. Lück
Journal: J. Amer. Math. Soc. 27 (2014), 339-388
MSC (2010): Primary 18F25, 19A31, 19B28, 19G24, 22E40, 57N99
DOI: https://doi.org/10.1090/S0894-0347-2014-00782-7
Published electronically: January 15, 2014
MathSciNet review: 3164984
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ G$ be a cocompact lattice in a virtually connected Lie group or the fundamental group of a three-dimensional manifold. We prove the $ K$- and $ L$-theoretic Farrell-Jones Conjectures for $ G$.


References [Enhancements On Off] (What's this?)

  • [1] Herbert Abels, Parallelizability of proper actions, global $ K$-slices and maximal compact subgroups, Math. Ann. 212 (1974/75), 1-19. MR 0375264 (51 #11460)
  • [2] Arthur Bartels, Siegfried Echterhoff, and Wolfgang Lück, Inheritance of isomorphism conjectures under colimits, $ K$-theory and noncommutative geometry, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2008, pp. 41-70. MR 2513332 (2012a:55005), https://doi.org/10.4171/060-1/2
  • [3] Arthur Bartels and Wolfgang Lück, Isomorphism conjecture for homotopy $ K$-theory and groups acting on trees, J. Pure Appl. Algebra 205 (2006), no. 3, 660-696. MR 2210223 (2007e:19005), https://doi.org/10.1016/j.jpaa.2005.07.020
  • [4] Arthur Bartels and Wolfgang Lück, Induction theorems and isomorphism conjectures for $ K$- and $ L$-theory, Forum Math. 19 (2007), no. 3, 379-406. MR 2328114 (2008j:19003), https://doi.org/10.1515/FORUM.2007.016
  • [5] Arthur Bartels and Wolfgang Lück, On crossed product rings with twisted involutions, their module categories and $ L$-theory, Cohomology of groups and algebraic $ K$-theory, Adv. Lect. Math. (ALM), vol. 12, Int. Press, Somerville, MA, 2010, pp. 1-54. MR 2655174 (2011m:19003)
  • [6] Arthur Bartels and Wolfgang Lück, The Borel conjecture for hyperbolic and $ {\rm CAT}(0)$-groups, Ann. of Math. (2) 175 (2012), no. 2, 631-689. MR 2993750, https://doi.org/10.4007/annals.2012.175.2.5
  • [7] A. Bartels and W. Lück, The Farrell-Hsiang method revisited, Math. Ann. 354 (2012), no. 1, 209-226. MR 2957625, https://doi.org/10.1007/s00208-011-0727-3
  • [8] Arthur Bartels, Wolfgang Lück, and Holger Reich, Equivariant covers for hyperbolic groups, Geom. Topol. 12 (2008), no. 3, 1799-1882. MR 2421141 (2009d:20102), https://doi.org/10.2140/gt.2008.12.1799
  • [9] Arthur Bartels, Wolfgang Lück, and Holger Reich, The $ K$-theoretic Farrell-Jones conjecture for hyperbolic groups, Invent. Math. 172 (2008), no. 1, 29-70. MR 2385666 (2009c:19002), https://doi.org/10.1007/s00222-007-0093-7
  • [10] Arthur Bartels, Wolfgang Lück, and Holger Reich, On the Farrell-Jones conjecture and its applications, J. Topol. 1 (2008), no. 1, 57-86. MR 2365652 (2008m:19001), https://doi.org/10.1112/jtopol/jtm008
  • [11] A. Bartels, W. Lück, and S. Weinberger, On hyperbolic groups with spheres as boundary. arXiv:0911.3725v1 [math.GT], to appear in the Journal of Differential Geometry (2009).
  • [12] Arthur Bartels and Holger Reich, Coefficients for the Farrell-Jones conjecture, Adv. Math. 209 (2007), no. 1, 337-362. MR 2294225 (2008a:19002), https://doi.org/10.1016/j.aim.2006.05.005
  • [13] Martin R. Bridson and André Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. MR 1744486 (2000k:53038)
  • [14] Kenneth S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York, 1982. MR 672956 (83k:20002)
  • [15] J. Bryant, S. Ferry, W. Mio, and S. Weinberger, Topology of homology manifolds, Ann. of Math. (2) 143 (1996), no. 3, 435-467. MR 1394965 (97b:57017), https://doi.org/10.2307/2118532
  • [16] Frank Connolly and Tadeusz Koźniewski, Rigidity and crystallographic groups. I, Invent. Math. 99 (1990), no. 1, 25-48. MR 1029389 (91g:57019), https://doi.org/10.1007/BF01234410
  • [17] James F. Davis, Qayum Khan, and Andrew Ranicki, Algebraic $ K$-theory over the infinite dihedral group: an algebraic approach, Algebr. Geom. Topol. 11 (2011), no. 4, 2391-2436. MR 2835234 (2012h:19008), https://doi.org/10.2140/agt.2011.11.2391
  • [18] James F. Davis, Frank Quinn, and Holger Reich, Algebraic $ K$-theory over the infinite dihedral group: a controlled topology approach, J. Topol. 4 (2011), no. 3, 505-528. MR 2832565 (2012g:19006), https://doi.org/10.1112/jtopol/jtr009
  • [19] Patrick B. Eberlein, Geometry of nonpositively curved manifolds, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1996. MR 1441541 (98h:53002)
  • [20] F. T. Farrell and W. C. Hsiang, The Whitehead group of poly-(finite or cyclic) groups, J. London Math. Soc. (2) 24 (1981), no. 2, 308-324. MR 631942 (83b:20041), https://doi.org/10.1112/jlms/s2-24.2.308
  • [21] F. T. Farrell and W. C. Hsiang, Topological characterization of flat and almost flat Riemannian manifolds $ M^{n}$ $ (n\not =3,\,4)$, Amer. J. Math. 105 (1983), no. 3, 641-672. MR 704219 (84k:57017), https://doi.org/10.2307/2374318
  • [22] F. T. Farrell and L. E. Jones, The surgery $ L$-groups of poly-(finite or cyclic) groups, Invent. Math. 91 (1988), no. 3, 559-586. MR 928498 (89d:57049), https://doi.org/10.1007/BF01388787
  • [23] F. T. Farrell and L. E. Jones, Isomorphism conjectures in algebraic $ K$-theory, J. Amer. Math. Soc. 6 (1993), no. 2, 249-297. MR 1179537 (93h:57032), https://doi.org/10.2307/2152801
  • [24] I. Hambleton, E. K. Pedersen, and D. Rosenthal, Assembly maps for group extensions in $ K$- and $ L$-theory. Preprint, arXiv:math.KT/0709.0437v1, to appear in Pure and Applied Mathematics Quarterly (2007).
  • [25] Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, vol. 80, Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, 1978. MR 514561 (80k:53081)
  • [26] Nigel Higson and Gennadi Kasparov, $ E$-theory and $ KK$-theory for groups which act properly and isometrically on Hilbert space, Invent. Math. 144 (2001), no. 1, 23-74. MR 1821144 (2002k:19005), https://doi.org/10.1007/s002220000118
  • [27] P. Kühl, Isomorphismusvermutungen und 3-Mannigfaltigkeiten. Preprint, arXiv:0907.0729v1 [math.KT] (2009).
  • [28] Wolfgang Lück, $ K$- and $ L$-theory of the semi-direct product of the discrete 3-dimensional Heisenberg group by $ {\mathbb{Z}}/4$, Geom. Topol. 9 (2005), 1639-1676 (electronic). MR 2175154 (2006e:19007), https://doi.org/10.2140/gt.2005.9.1639
  • [29] Wolfgang Lück, Survey on classifying spaces for families of subgroups, Infinite groups: geometric, combinatorial and dynamical aspects, Progr. Math., vol. 248, Birkhäuser, Basel, 2005, pp. 269-322. MR 2195456 (2006m:55036), https://doi.org/10.1007/3-7643-7447-0_7
  • [30] W. Lück and H. Reich, The Baum-Connes and the Farrell-Jones conjectures in $ K$- and $ L$-theory, In Handbook of $ K$-theory. Vol. 1, 2, Springer, Berlin, 2005, pp. 703-842.
  • [31] Frank Quinn, An obstruction to the resolution of homology manifolds, Michigan Math. J. 34 (1987), no. 2, 285-291. MR 894878 (88j:57016), https://doi.org/10.1307/mmj/1029003559
  • [32] Frank Quinn, Algebraic $ K$-theory over virtually abelian groups, J. Pure Appl. Algebra 216 (2012), no. 1, 170-183. MR 2826431 (2012f:19006), https://doi.org/10.1016/j.jpaa.2011.06.001
  • [33] Frank Quinn, Controlled K-theory I: Basic theory, Pure Appl. Math. Q. 8 (2012), no. 2, 329-421. MR 2900172
  • [34] M. S. Raghunathan, Discrete subgroups of Lie groups, Springer-Verlag, New York, 1972. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68. MR 0507234 (58 #22394a)
  • [35] S. K. Roushon, The Farrell-Jones isomorphism conjecture for 3-manifold groups, J. K-Theory 1 (2008), no. 1, 49-82. MR 2424566 (2010e:57025), https://doi.org/10.1017/is007011012jkt005
  • [36] Sayed K. Roushon, The isomorphism conjecture for 3-manifold groups and $ K$-theory of virtually poly-surface groups, J. K-Theory 1 (2008), no. 1, 83-93. MR 2424567 (2009m:57001), https://doi.org/10.1017/is007011012jkt006
  • [37] J.-P. Serre, A course in arithmetic, Springer-Verlag, New York, 1973. Translated from the French; Graduate Texts in Mathematics, No. 7. MR 0344216 (49 #8956)
  • [38] Lie groups and Lie algebras. II, Encyclopaedia of Mathematical Sciences, vol. 21, Springer-Verlag, Berlin, 2000. Discrete subgroups of Lie groups and cohomologies of Lie groups and Lie algebras; A translation of Current problems in mathematics. Fundamental directions. Vol. 21 (Russian), Akad. Nauk SSSR, Vsesoyuz. Inst.Nauchn. i Tekhn. Inform. (VINITI), Moscow, 1988 [ MR0968444 (89f:22001)]; Translated by John Danskin; Translation edited by A. L. Onishchik and E. B. Vinberg. MR 1756406 (2001a:22001)
  • [39] Christian Wegner, The $ K$-theoretic Farrell-Jones conjecture for CAT(0)-groups, Proc. Amer. Math. Soc. 140 (2012), no. 3, 779-793. MR 2869063, https://doi.org/10.1090/S0002-9939-2011-11150-X

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2010): 18F25, 19A31, 19B28, 19G24, 22E40, 57N99

Retrieve articles in all journals with MSC (2010): 18F25, 19A31, 19B28, 19G24, 22E40, 57N99


Additional Information

A. Bartels
Affiliation: Westfälische Wilhelms-Universität Münster, Mathematicians Institut,Einsteinium. 62, D-48149 Münster, Germany
Email: bartelsa@math.uni-muenster.de

F. T. Farrell
Affiliation: Department of Mathematics, Suny, Binghamton, New York, New York 13902
Email: farrell@math.binghamton.edu

W. Lück
Affiliation: Mathematicians Institut der Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany
Email: wolfgang.lueck@him.uni-bonn.de

DOI: https://doi.org/10.1090/S0894-0347-2014-00782-7
Keywords: Farrell-Jones Conjecture, $K$- and $L$-theory of group rings, cocompact lattices in virtually connected Lie groups, fundamental groups of $3$-manifolds.
Received by editor(s): January 3, 2011
Received by editor(s) in revised form: April 16, 2013
Published electronically: January 15, 2014
Article copyright: © Copyright 2014 American Mathematical Society

American Mathematical Society