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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 54 (12 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 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 50 (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.
THE FARRELLJONES 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 3dimensional manifold. We prove the K and Ltheoretic FarrellJones 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 3dimensional manifold. We prove the K and Ltheoretic FarrellJones Conjecture for G.
Algebraic Ktheory 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 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.
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 20 (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.
FINITENESS OBSTRUCTIONS AND EULER CHARACTERISTICS OF CATEGORIES
, 2009
"... We introduce notions of finiteness obstruction, Euler characteristic, L²Euler 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 L²Euler character ..."
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Cited by 14 (2 self)
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We introduce notions of finiteness obstruction, Euler characteristic, L²Euler 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 L²Euler characteristic are respectively its RΓrank and L²rank. We also extend the second author’s Ktheoretic Möbius inversion from finite categories to quasifinite 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 wordhyperbolic groups have finite asymptotic dimension. This is important in connection with our forthcoming proof of the FarrellJones conjecture for K∗(RG) for every wordhyperbolic 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 wordhyperbolic groups have finite asymptotic dimension. This is important in connection with our forthcoming proof of the FarrellJones conjecture for K∗(RG) for every wordhyperbolic group G and every coefficient ring R. 20F67; 37D40 1
On hyperbolic groups with spheres as boundary
, 2008
"... Let G be a torsionfree 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 torsionfree 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 Ktheory. http://www.arxiv.org
"... Abstract. This note explains consequences of recent work of Frank Quinn for computations of Nil groups in algebraic Ktheory, in particular the Nil groups occurring in the Ktheory of polynomial rings, Laurent polynomial rings, and the group ring of the infinite dihedral group. 1. Statement of Resul ..."
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Abstract. This note explains consequences of recent work of Frank Quinn for computations of Nil groups in algebraic Ktheory, in particular the Nil groups occurring in the Ktheory 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 Kgroup of Bass and Quillen. Bass defines the NKgroups NKq(R) = ker(KqR[t] → KqR) where the map on Kgroups is induced by the ring map R[t] → R, f(t) ↦ → f(0). The NKgroups are often called Nilgroups because they are related to nilpotent endomorphisms of projective Rmodules. Let G be a group. Let OrG be its the orbit category; objects are Gsets G/H where H is a subgroup of G and morphisms are Gmaps. DavisLück [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 GCWcomplex 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 Gactions with isotopy in F. It is characterized up to Ghomotopy type as a GCWcomplex 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.