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49
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 105 (31 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 51 (13 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 48 (13 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 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 40 (19 self)
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Abstract. We prove the Ktheoretic FarrellJones Conjecture for hyperbolic groups with (twisted) coefficients in any associative ring with unit.
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 24 (5 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.
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 15 (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.
ON THE KTHEORY OF GROUPS WITH FINITE ASYMPTOTIC DIMENSION
"... Abstract. It is proved that the assembly maps in algebraic K and Ltheory with respect to the family of finite subgroups is injective for groups Γ with finite asymptotic dimension that admit a finite model for EΓ. The result also applies to certain groups that admit only a finite dimensional model ..."
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Cited by 14 (2 self)
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Abstract. It is proved that the assembly maps in algebraic K and Ltheory with respect to the family of finite subgroups is injective for groups Γ with finite asymptotic dimension that admit a finite model for EΓ. The result also applies to certain groups that admit only a finite dimensional model for EΓ. In particular, it applies to discrete subgroups of virtually connected Lie groups.
Higher algebraic Ktheory (after Quillen, . . . )
, 2007
"... We give a short introduction (with a few proofs) to higher algebraic Ktheory (mainly of schemes) based on the work of Quillen, Waldhausen, Thomason and others. ..."
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Cited by 13 (0 self)
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We give a short introduction (with a few proofs) to higher algebraic Ktheory (mainly of schemes) based on the work of Quillen, Waldhausen, Thomason and others.
Induction theorems and isomorphism conjectures for K and Ltheory
 PREPRINTREIHE SFB 478 — 38 STRUKTUREN IN DER MATHEMATIK, HEFT 331
, 2004
"... The FarrellJones and the BaumConnes Conjecture say that one can compute the algebraic K and Ltheory of the group ring and the topological Ktheory of the reduced group C∗algebra of a group G in terms of these functors for the virtually cyclic subgroups or the finite subgroups of G. By induction ..."
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Cited by 13 (9 self)
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The FarrellJones and the BaumConnes Conjecture say that one can compute the algebraic K and Ltheory of the group ring and the topological Ktheory of the reduced group C∗algebra of a group G in terms of these functors for the virtually cyclic subgroups or the finite subgroups of G. By induction theory we want to reduce these families of subgroups to a smaller family, for instance to the family of subgroups which are either finite hyperelementary or extensions of finite hyperelementary groups with Z as kernel or to the family of finite cyclic subgroups. Roughly speaking, we extend the induction theorems of Dress for finite groups to infinite groups.
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 13 (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