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48
Isomorphism conjectures in algebraic Ktheory
 J. Amer. Math. Soc
, 1993
"... 1.1 The Isomorphism Conjecture in algebraic Ktheory........ 2 1.2 Main Results and Corollaries.................... 4 1.3 A brief outline............................ 6 ..."
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Cited by 109 (12 self)
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1.1 The Isomorphism Conjecture in algebraic Ktheory........ 2 1.2 Main Results and Corollaries.................... 4 1.3 A brief outline............................ 6
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.
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.
Algebraic Ktheory over the infinite dihedral group, submitted
 URL http://arxiv.org/abs/0803.1639
"... Abstract. A group G with an epimorphism G → D ∞ onto the infinite dihedral group D ∞ = Z2 ∗ Z2 = Z ⋊ Z2 inherits an amalgamated free product structure G = G1 ∗F G2 with F an index 2 subgroup of G1 and G2. Also, there is an index 2 subgroup ¯ G ⊂ G with an HNN structure ¯ G = F ⋊α Z. For such a G we ..."
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Cited by 13 (2 self)
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Abstract. A group G with an epimorphism G → D ∞ onto the infinite dihedral group D ∞ = Z2 ∗ Z2 = Z ⋊ Z2 inherits an amalgamated free product structure G = G1 ∗F G2 with F an index 2 subgroup of G1 and G2. Also, there is an index 2 subgroup ¯ G ⊂ G with an HNN structure ¯ G = F ⋊α Z. For such a G we obtain an isomorphism of reduced Nilgroups fNil∗(R[F]; R[G1 − F],R[G2 − F]) ∼ = Nil∗(R[F], f α) for any ring R. We use this to show that for any group Γ, there is an isomorphism H Γ n(EfbcΓ;KR) ∼ = H Γ n(EvcΓ;KR), which sharpens the Farrell–Jones isomorphism conjecture in algebraic Ktheory.
K– and L–theory of the semidirect product of the discrete 3–dimensional Heisenberg group by Z/4
, 2005
"... We compute the group homology, the topological K–theory of the reduced C ∗ – algebra, the algebraic K–theory and the algebraic L–theory of the group ring of the semidirect product of the threedimensional discrete Heisenberg group by Z/4. These computations will follow from the more general treat ..."
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Cited by 12 (1 self)
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We compute the group homology, the topological K–theory of the reduced C ∗ – algebra, the algebraic K–theory and the algebraic L–theory of the group ring of the semidirect product of the threedimensional discrete Heisenberg group by Z/4. These computations will follow from the more general treatment of a certain class of groups G which occur as extensions 1 → K → G → Q → 1 of a torsionfree group K by a group Q which satisfies certain assumptions. The key ingredients are the Baum–Connes and Farrell–Jones Conjectures and methods from equivariant algebraic topology.
Inheritance of isomorphism conjectures under colimits
, 2007
"... Abstract. We investigate when Isomorphism Conjectures, such as the ones due to BaumConnes, Bost and FarrellJones, are stable under colimits of groups over directed sets (with not necessarily injective structure maps). We show in particular that both the Ktheoretic FarrellJones Conjecture and the ..."
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Cited by 11 (9 self)
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Abstract. We investigate when Isomorphism Conjectures, such as the ones due to BaumConnes, Bost and FarrellJones, are stable under colimits of groups over directed sets (with not necessarily injective structure maps). We show in particular that both the Ktheoretic FarrellJones Conjecture and the Bost Conjecture with coefficients hold for those groups for which Higson, Lafforgue and Skandalis have disproved the BaumConnes Conjecture with coefficients.
The FarrellJones isomorphism conjecture for finite covolume hyperbolic actions and the algebraic Ktheory of Bianchi groups
"... Abstract. We prove the FarrellJones Isomorphism Conjecture for groups acting on complete hyperbolic manifolds with finite volume orbit space. We then apply this result to show that for any Bianchi group Γ, W h(Γ), ˜ K0(ZΓ), and Ki(ZΓ) vanish for i ≤ −1. 1. ..."
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Cited by 10 (2 self)
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Abstract. We prove the FarrellJones Isomorphism Conjecture for groups acting on complete hyperbolic manifolds with finite volume orbit space. We then apply this result to show that for any Bianchi group Γ, W h(Γ), ˜ K0(ZΓ), and Ki(ZΓ) vanish for i ≤ −1. 1.
Splitting with Continuous Control in Algebraic Ktheory
, 2002
"... Abstract. In this work, the continuously controlled assembly map in algebraic Ktheory, as developed by Carlsson and Pedersen, is proved to be a split injection for groups Γ that satisfy certain geometric conditions. The group Γ is allowed to have torsion, generalizing a result of Carlsson and Peder ..."
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Cited by 10 (5 self)
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Abstract. In this work, the continuously controlled assembly map in algebraic Ktheory, as developed by Carlsson and Pedersen, is proved to be a split injection for groups Γ that satisfy certain geometric conditions. The group Γ is allowed to have torsion, generalizing a result of Carlsson and Pedersen. Combining this with a result of John Moody, K0(kΓ) is proved to be isomorphic to the colimit of K0(kH) over the finite subgroups H of Γ, when Γ is a virtually polycyclic group and k is a field of characteristic zero. 1.
The BaumConnes conjecture via localization of categories
"... Abstract. We redefine the BaumConnes assembly map using simplicial approximation ..."
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Cited by 9 (3 self)
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Abstract. We redefine the BaumConnes assembly map using simplicial approximation