K - and L -theory of group rings over G L n ( 𝐙 )
Bartels, Arthur1 ; Lück, Wolfgang2 ; Reich, Holger3 ; Rüping, Henrik2
1 Mathematisches Institut, Westfälische Wilhelms-Universität Münster Einsteinstr. 60 48149 Münster Germany
2 Mathematisches Institut, Rheinische Wilhelms-Universität Bonn Endenicher Allee 60 53115 Bonn Germany
3 Institut für Mathematik, Freie Universität Berlin Arnimallee 7 14195 Berlin Germany
Publications Mathématiques de l'IHÉS, Tome 119 (2014), pp. 97-125

We prove the K - and L -theoretic Farrell-Jones Conjecture (with coefficients in additive categories) for G L n ( 𝐙 )

DOI : 10.1007/s10240-013-0055-0
Keywords: Abelian Group, Volume Function, Group Ring, Cyclic Subgroup, Wreath Product

Bartels, Arthur 1 ; Lück, Wolfgang 2 ; Reich, Holger 3 ; Rüping, Henrik 2

1 Mathematisches Institut, Westfälische Wilhelms-Universität Münster Einsteinstr. 60 48149 Münster Germany
2 Mathematisches Institut, Rheinische Wilhelms-Universität Bonn Endenicher Allee 60 53115 Bonn Germany
3 Institut für Mathematik, Freie Universität Berlin Arnimallee 7 14195 Berlin Germany 
@article{PMIHES_2014__119__97_0,
     author = {Bartels, Arthur and L\"uck, Wolfgang and Reich, Holger and R\"uping, Henrik},
     title = {$K$- and \protect\emph{$L$}-theory of group rings over $GL_n ( \mathbf{Z} )$},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {97--125},
     year = {2014},
     publisher = {Springer Berlin Heidelberg},
     address = {Berlin/Heidelberg},
     volume = {119},
     doi = {10.1007/s10240-013-0055-0},
     language = {en},
     url = {https://www.numdam.org/articles/10.1007/s10240-013-0055-0/}
}
TY  - JOUR
AU  - Bartels, Arthur
AU  - Lück, Wolfgang
AU  - Reich, Holger
AU  - Rüping, Henrik
TI  - $K$- and $L$-theory of group rings over $GL_n ( \mathbf{Z} )$
JO  - Publications Mathématiques de l'IHÉS
PY  - 2014
SP  - 97
EP  - 125
VL  - 119
PB  - Springer Berlin Heidelberg
PP  - Berlin/Heidelberg
UR  - https://www.numdam.org/articles/10.1007/s10240-013-0055-0/
DO  - 10.1007/s10240-013-0055-0
LA  - en
ID  - PMIHES_2014__119__97_0
ER  - 
%0 Journal Article
%A Bartels, Arthur
%A Lück, Wolfgang
%A Reich, Holger
%A Rüping, Henrik
%T $K$- and $L$-theory of group rings over $GL_n ( \mathbf{Z} )$
%J Publications Mathématiques de l'IHÉS
%D 2014
%P 97-125
%V 119
%I Springer Berlin Heidelberg
%C Berlin/Heidelberg
%U https://www.numdam.org/articles/10.1007/s10240-013-0055-0/
%R 10.1007/s10240-013-0055-0
%G en
%F PMIHES_2014__119__97_0
Bartels, Arthur; Lück, Wolfgang; Reich, Holger; Rüping, Henrik. $K$- and $L$-theory of group rings over $GL_n ( \mathbf{Z} )$. Publications Mathématiques de l'IHÉS, Tome 119 (2014), pp. 97-125. doi: 10.1007/s10240-013-0055-0

[1.] A. Bartels, T. Farrell, and W. Lück, The Farrell-Jones conjecture for cocompact lattices in virtually connected Lie groups. | arXiv

[2.] Bartels, A.; Lück, W. On twisted group rings with twisted involutions, their module categories and L-theory, Cohomology of Groups and Algebraic K-Theory (2009), pp. 1-55 | MR | Zbl

[3.] Bartels, A.; Lück, W. Geodesic flow for CAT(0)-groups, Geom. Topol., Volume 16 (2012), pp. 1345-1391 | MR | Zbl | DOI

[4.] Bartels, A.; Lück, W. The Borel conjecture for hyperbolic and CAT(0)-groups, Ann. Math. (2), Volume 175 (2012), pp. 631-689 | MR | Zbl | DOI

[5.] Bartels, A.; Lück, W.; Reich, H. The K-theoretic Farrell-Jones conjecture for hyperbolic groups, Invent. Math., Volume 172 (2008), pp. 29-70 | MR | Zbl | DOI

[6.] Bartels, A.; Lück, W.; Reich, H. On the Farrell-Jones conjecture and its applications, Topology, Volume 1 (2008), pp. 57-86 | MR | Zbl | DOI

[7.] Bartels, A.; Reich, H. Coefficients for the Farrell-Jones conjecture, Adv. Math., Volume 209 (2007), pp. 337-362 | MR | Zbl | DOI

[8.] Bridson, M. R.; Haefliger, A. Metric Spaces of Non-Positive Curvature (1999) | MR | Zbl

[9.] Brown, K. S. Cohomology of Groups (1982) | MR | Zbl

[10.] Davis, J. F.; Quinn, F.; Reich, H. Algebraic K-theory over the infinite dihedral group: a controlled topology approach, Topology, Volume 4 (2011), pp. 505-528 | MR | Zbl | DOI

[11.] Dixon, J. D.; Mortimer, B. Permutation Groups (1996) | MR | Zbl

[12.] Eberlein, P. B. Geometry of Nonpositively Curved Manifolds (1996) | MR | Zbl

[13.] Farrell, F. T.; Jones, L. E. Isomorphism conjectures in algebraic K-theory, J. Am. Math. Soc., Volume 6 (1993), pp. 249-297 | MR | Zbl

[14.] Farrell, F. T.; Jones, L. E. Rigidity for aspherical manifolds with π1⊂GLm(R), Asian J. Math., Volume 2 (1998), pp. 215-262 | MR | Zbl

[15.] Farrell, F. T.; Roushon, S. K. The Whitehead groups of braid groups vanish, Int. Math. Res. Not., Volume 10 (2000), pp. 515-526 | MR | Zbl | DOI

[16.] Grayson, D. R. Reduction theory using semistability, Comment. Math. Helv., Volume 59 (1984), pp. 600-634 | MR | Zbl | DOI

[17.] Griffiths, P.; Harris, J. Principles of Algebraic Geometry (1978) | Zbl | MR

[18.] Helgason, S. Differential Geometry, Lie Groups, and Symmetric Spaces (1978) | MR | Zbl

[19.] P. Kühl, Isomorphismusvermutungen und 3-Mannigfaltigkeiten. Preprint, | arXiv

[20.] Lück, W. Survey on classifying spaces for families of subgroups, Infinite Groups: Geometric, Combinatorial and Dynamical Aspects (2005), pp. 269-322 | MR | Zbl

[21.] Lück, W.; Reich, H. The Baum-Connes and the Farrell-Jones conjectures in K- and L-theory, Handbook of K-Theory, vols. 1, 2 (2005), pp. 703-842 | MR | Zbl

[22.] Neukirch, J. Algebraic Number Theory (1999) | MR | Zbl

[23.] Roushon, S. K. The Farrell-Jones isomorphism conjecture for 3-manifold groups, K-Theory, Volume 1 (2008), pp. 49-82 | MR | Zbl

[24.] Wegner, C. The K-theoretic Farrell-Jones conjecture for CAT(0)-groups, Proc. Am. Math. Soc., Volume 140 (2012), pp. 779-793 | MR | Zbl | DOI

Cité par Sources :