Related papers: Relating the Farrell Nil-groups to the Waldhausen …

An open question is whether the map $\widetilde{K_0 }\mathbb{Z} G \rightarrow \widetilde{K_0 }\mathbb{Q} G$ in reduced $K$-theory from the integral to the rational group ring is trivial for any group $G$. We will show that this is false,…

Using a theorem of L\"uck-Reich-Rognes-Varisco, we show that the Whitehead group of Thompson's group T is infinitely generated, even when tensored with the rationals. To this end we describe the structure of the centralizers and normalizers…

We prove that for any two continuous minimal (topologically free) actions of the infinite dihedral group on an infinite compact Hausdorff space, they are continuously orbit equivalent only if they are conjugate. We also show the above fails…

Let G be a cocompact lattice in a virtually connected Lie group or the fundamental group of a 3-manifold. We prove the K-theoretic Farrell-Jones Conjecture (up to dimension one) and the L-theoretic Farrell-Jones Conjecture for G, where we…

We study the Fibered Isomorphism conjecture of Farrell and Jones for groups acting on trees. We show that under certain conditions the conjecture is true for groups acting on trees when the stabilizers satisfy the conjecture. These…

If G is a finite group and k is a field, there is a natural construction of a Hopf algebra over k associated to G, the Drinfel'd double D(G). We prove that if G is any finite real reflection group with Drinfel'd double D(G) over an…

In this article, we generalize Loday and Pirashvili's [10] computation of the Ext-category of Leibniz bimodules for a simple Lie algebra to the case of a simple (non Lie) Leibniz algebra. Most of the arguments generalize easily, while the…

Hans Zassenhaus conjectured that every torsion unit of the integral group ring of a finite group $G$ is conjugate within the rational group algebra to an element of the form $\pm g$ with $g\in G$. This conjecture has been disproved recently…

We investigate the relative assembly map from the family of finite subgroups to the family of virtually cyclic subgroups for the algebraic $K$-theory of twisted group rings of a group G with coefficients in a regular ring R or, more…

Walker's cancellation theorem says that if B+Z is isomorphic to C+Z in the category of abelian groups, then B is isomorphic to C. We construct an example in a diagram category of abelian groups where the theorem fails. As a consequence, the…

We prove that the shifted vanishing cycles and nearby cycles commute with Verdier dualizing up to a {\bf natural} isomorphism, even when the coefficients are not in a field.

We apply categorical machinery to the problem of defining cyclic cohomology with coefficients in two particular cases, namely quasi-Hopf algebras and Hopf algebroids. In the case of the former, no definition was thus far available in the…

We extend the main vanishing theorem in a paper of de Fernex and Ein to singular varieties without assuming locally complete intersection.

A proof based on reduction to finite fields of Esnault-Viehweg's stronger version of Sommese Vanishing Theorem for $k$-ample line bundles is given. This result is used to give different proofs of isotriviality results of A. Parshin and L.…

We prove that if $X \to Y$ is a (geometrically) regular morphism of Noetherian schemes, then from a Nisnevich-local perspective, the Gersten complex for Quillen $K$-theory on $X$ becomes acyclic in degrees beyond the Krull dimension of $Y$.…

Let G be a Frobenius group with the Frobenius kernel K. Suppose that G contains a nontrival subgroup D \subseteq K such that the normalizer N_G(D) \not\subseteq K. When D is no 2-group, Flavell proved, without using character theory, that K…

We prove the K- and L-theoretic Farrell-Jones Conjecture (with coefficients in additive categories) for GL_n(Z).

We consider (random) walks in a multidimensional orthant. Using the idea of universality in probability theory, one can associate a unique polyhedral domain to any given walk model. We use this connection to prove two sets of new results.…

Let N be a simply connected, connected real nilpotent Lie group of finite dimension n. We study subgroups $\Gamma$ in $\Aff (N)=N\rtimes \Aut (N)$ acting properly discontinuously and cocompactly on N. This situation is a natural…

We first prove that the Whitehead group of a torsion-free virtually solvable linear group vanishes. Next we make a reduction of the fibered isomorphism conjecture from virtually solvable groups to a class of virtually solvable Q-linear…