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40
Coefficients for the Farrell-Jones conjecture
- Preprintreihe SFB 478 — Geometrische Strukturen in der Mathematik, Heft 402
"... Abstract. We introduce the Farrell-Jones Conjecture with coefficients in an additive category with G-action. This is a variant of the Farrell-Jones Conjecture about the algebraic K- or L-Theory 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 54 (12 self)
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Abstract. We introduce the Farrell-Jones Conjecture with coefficients in an additive category with G-action. This is a variant of the Farrell-Jones Conjecture about the algebraic K- or L-Theory 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 set-up with coefficients we obtain new results about the original Farrell-Jones Conjecture. The conjecture with coefficients implies the fibered version of the Farrell-Jones Conjecture. 1.
The Borel conjecture for hyperbolic and CAT(0)-groups
- ANN. OF MATH
, 2009
"... 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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Cited by 50 (12 self)
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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.
THE FARRELL-JONES CONJECTURE FOR COCOMPACT LATTICES IN VIRTUALLY CONNECTED LIE GROUPS
, 2013
"... Let G be a cocompact lattice in a virtually connected Lie group or the fundamental group of a 3-dimensional manifold. We prove the K- and L-theoretic Farrell-Jones Conjecture for G. ..."
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Cited by 25 (4 self)
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Let G be a cocompact lattice in a virtually connected Lie group or the fundamental group of a 3-dimensional manifold. We prove the K- and L-theoretic Farrell-Jones Conjecture for G.
Algebraic K-theory over the infinite dihedral group
, 2008
"... 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 ..."
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Cited by 23 (4 self)
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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 Nil-groups 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 K-theory.
Inheritance of isomorphism conjectures under colimits
, 2007
"... Abstract. We investigate when Isomorphism Conjectures, such as the ones due to Baum-Connes, Bost and Farrell-Jones, are stable under colimits of groups over directed sets (with not necessarily injective structure maps). We show in particular that both the K-theoretic Farrell-Jones Conjecture and the ..."
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Cited by 20 (9 self)
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Abstract. We investigate when Isomorphism Conjectures, such as the ones due to Baum-Connes, Bost and Farrell-Jones, are stable under colimits of groups over directed sets (with not necessarily injective structure maps). We show in particular that both the K-theoretic Farrell-Jones Conjecture and the Bost Conjecture with coefficients hold for those groups for which Higson, Lafforgue and Skandalis have disproved the Baum-Connes Conjecture with coefficients.
FINITENESS OBSTRUCTIONS AND EULER CHARACTERISTICS OF CATEGORIES
, 2009
"... We introduce notions of finiteness obstruction, Euler characteristic, L²-Euler characteristic, and Möbius inversion for wide classes of categories. The finiteness obstruction of a category Γ of type (FP) is a class in the projective class group K0(RΓ); the Euler characteristic and L²-Euler character ..."
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Cited by 14 (2 self)
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We introduce notions of finiteness obstruction, Euler characteristic, L²-Euler characteristic, and Möbius inversion for wide classes of categories. The finiteness obstruction of a category Γ of type (FP) is a class in the projective class group K0(RΓ); the Euler characteristic and L²-Euler characteristic are respectively its RΓ-rank and L²-rank. We also extend the second author’s K-theoretic Möbius inversion from finite categories to quasi-finite categories. Our main
Equivariant covers of hyperbolic groups
- Preprintreihe SFB 478 — Geometrische Strukturen in der Mathematik, Heft 434
, 2006
"... We prove an equivariant version of the fact that word-hyperbolic groups have finite asymptotic dimension. This is important in connection with our forthcoming proof of the Farrell-Jones conjecture for K∗(RG) for every word-hyperbolic group G and every coefficient ring R. 20F67; 37D40 1 ..."
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Cited by 14 (6 self)
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We prove an equivariant version of the fact that word-hyperbolic groups have finite asymptotic dimension. This is important in connection with our forthcoming proof of the Farrell-Jones conjecture for K∗(RG) for every word-hyperbolic group G and every coefficient ring R. 20F67; 37D40 1
On hyperbolic groups with spheres as boundary
, 2008
"... Let G be a torsion-free hyperbolic group and let n ≥ 6 be an integer. We prove that G is the fundamental group of a closed aspherical manifold if the boundary of G is homeomorphic to an (n−1)-dimensional sphere. ..."
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Cited by 11 (5 self)
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Let G be a torsion-free hyperbolic group and let n ≥ 6 be an integer. We prove that G is the fundamental group of a closed aspherical manifold if the boundary of G is homeomorphic to an (n−1)-dimensional sphere.
Some remarks on Nil groups in algebraic K-theory. http://www.arxiv.org
"... Abstract. This note explains consequences of recent work of Frank Quinn for computations of Nil groups in algebraic K-theory, in particular the Nil groups occurring in the K-theory of polynomial rings, Laurent polynomial rings, and the group ring of the infinite dihedral group. 1. Statement of Resul ..."
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Cited by 8 (0 self)
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Abstract. This note explains consequences of recent work of Frank Quinn for computations of Nil groups in algebraic K-theory, in particular the Nil groups occurring in the K-theory of polynomial rings, Laurent polynomial rings, and the group ring of the infinite dihedral group. 1. Statement of Results Let R be a ring with unit. For an integer q, let KqR be the algebraic K-group of Bass and Quillen. Bass defines the NK-groups NKq(R) = ker(KqR[t] → KqR) where the map on K-groups is induced by the ring map R[t] → R, f(t) ↦ → f(0). The NK-groups are often called Nil-groups because they are related to nilpotent endomorphisms of projective R-modules. Let G be a group. Let OrG be its the orbit category; objects are G-sets G/H where H is a subgroup of G and morphisms are G-maps. Davis-Lück [8] define a functor K: OrG → Spectra with the key property πqK(G/H) = Kq(RH). The utility of such a functor is to allow the definition of an equivariant homology theory, indeed for a G-CW-complex X, one defines H G q (X;K) = πq(map G(−, X)+ ∧Or G K(−)) (see [8, section 4 and 7] for basic properties). Note that mapG(G/H, X) = XH is the fixed point functor and that the “coefficients ” of the homology theory are given by HG q (G/H;K) = Kq(RH). A family F of subgroups of G is a nonempty set of subgroups closed under subgroups and conjugation. For such a family, EF (short for EFG) is the classifying space for G-actions with isotopy in F. It is characterized up to G-homotopy type as a G-CW-complex so that EH F is contractible for subgroups H ∈ F and is empty for subgroups H ∈ F. Partially supported by a grant from the National Science Foundation.
