Holger Reich

The K -theoretic Farrell–Jones conjecture for hyperbolic groups

We prove the K-theoretic Farrell–Jones conjecture for hyperbolic groups with (twisted) coefficients in any associative ring with unit.

On the Isomorphism Conjecture in algebraic K -theory

The Isomorphism Conjecture is a conceptional approach towards a calculation of the algebraic K-theory of a group ring RΓ, where Γ is an infinite group. In this paper we prove the conjecture in dimensions n<2 for fundamental groups of closed Riemannian manifolds with strictly negative sectional curvature and arbitrary coefficient rings R. If R is regular this leads to a concrete calculation of low dimensional K-theory groups of RΓ in terms of the K-theory of R and the homology of the group.

On the Farrell-Jones conjecture for higher algebraic -theory

Here BΓ is the classifying space of the group Γ, and we denote by K−∞(R) the non-connective algebraic K-theory spectrum of the ring R. The homotopy groups of this spectrum are denoted Kn(R) and coincide with Quillen’s algebraicK-groups of R [Qui73] in positive dimensions and with the negative K-groups of Bass [Bas68] in negative dimensions. The homotopy groups of the spectrum X+∧K−∞(R) are denotedHn(X ;K−∞(R)). They yield a generalized homology theory and, in particular, standard computational tools such as the Atiyah-Hirzebruch spectral sequence apply to the left-hand side of the assembly map above. As a corollary of the main result of this paper we prove Conjecture 1.1 in the case where Γ is the fundamental group of a closed Riemannian manifold with strictly negative sectional curvature. In fact our result is more general and applies to group rings RΓ, where R is a completely arbitrary coefficient ring. Note that if one replaces in Conjecture 1.1 the coefficient ring Z by an arbitrary coefficient ringR the corresponding conjecture would be false already in the simplest non-trivial case: if Γ = C is the infinite cyclic group the Bass-Heller-Swan formula [BHS64], [Gra76, p. 236] for Kn(RC) = Kn(R[t±1]) yields that Kn(RC) ∼= Kn−1(R)⊕Kn(R)⊕NKn(R)⊕NKn(R), where NKn(R) is defined as the cokernel of the split inclusion Kn(R) → Kn(R[t]) and does not vanish in general. But since S is a model for BC one obtains on the left-hand side of the assembly map only Hn(BC;K−∞(R)) ∼= Kn(R)⊕Kn−1(R).

Algebraic K-theory over the infinite dihedral group: a controlled topology approach

We use controlled topology applied to the action of the infinite dihedral group on a partially compactified plane and deduce two consequences for algebraic K-theory. The first is that the family in the K-theoretic Farrell–Jones conjecture can be reduced to only those virtually cyclic groups that admit a surjection with finite kernel onto a cyclic group. The second is that the Waldhausen Nil groups for a group that maps epimorphically onto the infinite dihedral group can be computed in terms of the Farrell–Bass Nil groups of the index 2 subgroup that maps surjectively to the infinite cyclic group.

On the K – and L – Theory of the Algebra of Operators Affiliated to a Finite von Neumann Algebra

We construct a real valued dimension for arbitrary modules over the algebra of operators affiliated to a finite von Neumann algebra. Moreover we determine the algebraic K0and K1-group and the Lgroups of such an algebra.

Detecting K-Theory by Cyclic Homology

We discuss which part of the rationalized algebraic

Commuting Homotopy Limits and Smash Products

K

Algebraic K-theory of group rings and the cyclotomic trace map

-theory of a group ring is detected via trace maps to Hochschild homology, cyclic homology, periodic cyclic or negative cyclic homology.

On the Adams isomorphism for equivariant orthogonal spectra

In general the processes of taking a homotopy inverse limit of a diagram of spectra and smashing spectra with a fixed space do not commute. In this paper we investigate under what additional assumptions these two processes do commute. In fact we deal with an equivariant generalization which involves spectra and smash products over the orbit category of a discrete group. Such a situation naturally occurs if one studies the equivariant homology theory associated to topological cyclic homology. The main theorem of this paper will play a role in the generalization of the results obtained by Boekstedt, Hsiang and Madsen about the algebraic K-theory Novikov Conjecture to the assembly map for the family of virtually cyclic subgroups.

Assembly maps for topological cyclic homology of group algebras

We prove that the Farrell-Jones assembly map for connective alge- braic K-theory is rationally injective, under mild homological niteness condi- tions on the group and assuming that a weak version of the Leopoldt-Schneider conjecture holds for cyclotomic elds. This generalizes a result of Bokstedt, Hsiang, and Madsen, and leads to a concrete description of a large direct summand of Kn(Z(G)) Z Q in terms of group homology. In many cases the number theoretic conjectures are true, so we obtain rational injectivity results about assembly maps, in particular for Whitehead groups, under niteness assumptions on the group only. The proof uses the cyclotomic trace map to topological cyclic homology, Bokstedt-Hsiang-Madsens functor C, and new general injectivity results about the assembly maps for topological Hochschild homology and C.