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107
The type of the classifying space for a family of subgroups
 J. Pure Appl. Algebra
"... We define for a topological group G and a family of subgroupsF two versions for the classifying space for the family F, the GCWversion EF(G) and the numerable Gspace version JF(G). They agree if G is discrete, or if G is a Lie group and each element inF compact, or ifF is the family of compact su ..."
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Cited by 55 (28 self)
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We define for a topological group G and a family of subgroupsF two versions for the classifying space for the family F, the GCWversion EF(G) and the numerable Gspace version JF(G). They agree if G is discrete, or if G is a Lie group and each element inF compact, or ifF is the family of compact subgroups. We discuss special geometric models for these spaces for the family of compact open groups in special cases such as almost connected groups G and word hyperbolic groups G. We deal with the question whether there are finite models, models of finite type, finite dimensional models. We also discuss the relevance of these spaces for the BaumConnes Conjecture about the topological Ktheory of the reduced group C ∗algebra, for the FarrellJones Conjecture about the algebraic Kand Ltheory of group rings, for Completion Theorems and for classifying spaces for equivariant vector bundles and for other situations.
Spaces over a Category and Assembly Maps in Isomorphism Conjectures in Kand LTheory
"... : We give a unified approach to the Isomorphism Conjecture of Farrell and Jones on the algebraic K and Ltheory of integral group rings and to the BaumConnes Conjecture on the topological Ktheory of reduced group C algebras. The approach is through spectra over the orbit category of a discrete ..."
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Cited by 49 (12 self)
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: We give a unified approach to the Isomorphism Conjecture of Farrell and Jones on the algebraic K and Ltheory of integral group rings and to the BaumConnes Conjecture on the topological Ktheory of reduced group C algebras. The approach is through spectra over the orbit category of a discrete group G. We give several points of view on the assembly map for a family of subgroups and describe such assembly maps by a universal property generalizing the results of Weiss and Williams to the equivariant setting. The main tools are spaces and spectra over a category and the study of the associated generalized homology and cohomology theories and homotopy limits. Key words: Algebraic K and Ltheory, BaumConnes Conjecture, assembly maps, spaces and spectra over a category AMSclassification number: 57 Glen Bredon [5] introduced the orbit category Or(G) of a group G. Objects are homogeneous spaces G=H, considered as left Gsets, and morphisms are Gmaps. This is a useful construct for o...
The BaumConnes and the FarrellJones conjectures in K and Ltheory
 Preprintreihe SFB 478 — Geometrische Strukturen in der Mathematik, Heft 324
, 2004
"... Summary. We give a survey of the meaning, status and applications of the BaumConnes Conjecture about the topological Ktheory of the reduced group C ∗algebra and the FarrellJones Conjecture about the algebraic K and Ltheory of the group ring of a (discrete) group G. Key words: K and Lgroups o ..."
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Cited by 32 (24 self)
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Summary. We give a survey of the meaning, status and applications of the BaumConnes Conjecture about the topological Ktheory of the reduced group C ∗algebra and the FarrellJones Conjecture about the algebraic K and Ltheory of the group ring of a (discrete) group G. Key words: K and Lgroups of group rings and group C ∗algebras, BaumConnes
Isomorphism conjecture for homotopy Ktheory and groups acting on trees
, 2008
"... We discuss an analogon to the FarrellJones Conjecture for homotopy algebraic Ktheory. In particular, we prove that if a group G acts on a tree and all isotropy groups satisfy this conjecture, then G satisfies this conjecture. This result can be used to get rational injectivity results for the asse ..."
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Cited by 28 (13 self)
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We discuss an analogon to the FarrellJones Conjecture for homotopy algebraic Ktheory. In particular, we prove that if a group G acts on a tree and all isotropy groups satisfy this conjecture, then G satisfies this conjecture. This result can be used to get rational injectivity results for the assembly map in the FarrellJones Conjecture in algebraic Ktheory.
Coefficients for the FarrellJones conjecture
 Preprintreihe SFB 478 — Geometrische Strukturen in der Mathematik, Heft 402
"... Abstract. We introduce the FarrellJones Conjecture with coefficients in an additive category with Gaction. This is a variant of the FarrellJones Conjecture about the algebraic K or LTheory of a group ring RG. It allows to treat twisted group rings and crossed product rings. The conjecture with ..."
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Cited by 28 (11 self)
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Abstract. We introduce the FarrellJones Conjecture with coefficients in an additive category with Gaction. This is a variant of the FarrellJones Conjecture about the algebraic K or LTheory of a group ring RG. It allows to treat twisted group rings and crossed product rings. The conjecture with coefficients is stronger than the original conjecture but it has better inheritance properties. Since known proofs using controlled algebra carry over to the setup with coefficients we obtain new results about the original FarrellJones Conjecture. The conjecture with coefficients implies the fibered version of the FarrellJones Conjecture. 1.
The Ktheoretic FarrellJones Conjecture for hyperbolic groups
 Invent. Math
"... Abstract. We prove the Ktheoretic FarrellJones Conjecture for hyperbolic groups with (twisted) coefficients in any associative ring with unit. ..."
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Cited by 24 (17 self)
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Abstract. We prove the Ktheoretic FarrellJones Conjecture for hyperbolic groups with (twisted) coefficients in any associative ring with unit.
Periodic complexes and group actions
, 2001
"... In this paper we show that the cohomology of a connected CW–complex is periodic if and only if it is the base space of a spherical fibration with total space that is homotopically finite dimensional. As applications we characterize those discrete groups that act freely and properly on R n × S m; we ..."
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Cited by 18 (0 self)
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In this paper we show that the cohomology of a connected CW–complex is periodic if and only if it is the base space of a spherical fibration with total space that is homotopically finite dimensional. As applications we characterize those discrete groups that act freely and properly on R n × S m; we construct non–standard free actions of rank two simple groups on finite complexes Y ≃ S n × S m; and we prove that a finite p–group P acts freely on such a complex if and only if it does not contain a subgroup isomorphic to (Z/p)³.
L 2 determinant class and approximation of L 2 Betti numbers
 Trans. Amer. Math. Soc
"... A standing conjecture in L 2cohomology is that every finite CWcomplex X is of L 2determinant class. In this paper, we prove this whenever the fundamental group belongs to a large class G of groups containing e.g. all extensions of residually finite groups with amenable quotients, all residually am ..."
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Cited by 17 (3 self)
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A standing conjecture in L 2cohomology is that every finite CWcomplex X is of L 2determinant class. In this paper, we prove this whenever the fundamental group belongs to a large class G of groups containing e.g. all extensions of residually finite groups with amenable quotients, all residually amenable groups and free products of these. If, in addition, X is L 2acyclic, we also prove that the L 2determinant is a homotopy invariant. Even in the known cases, our proof of homotopy invariance is much shorter and easier than the previous ones. Under suitable conditions we give new approximation formulas for L 2Betti numbers.
The Borel conjecture for hyperbolic and CAT(0)groups
 ANN. OF MATH
, 2009
"... We prove the Borel Conjecture for a class of groups containing wordhyperbolic groups and groups acting properly, isometrically and cocompactly on a finite dimensional CAT(0)space. ..."
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Cited by 16 (12 self)
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We prove the Borel Conjecture for a class of groups containing wordhyperbolic groups and groups acting properly, isometrically and cocompactly on a finite dimensional CAT(0)space.