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

We prove that the Farrell-Jones assembly map for connective algebraic K-theory is rationally injective, under mild homological finiteness conditions on the group and assuming that a weak version of the Leopoldt-Schneider conjecture holds for cyclotomic fields. This generalizes a result of B\"okstedt, Hsiang, and Madsen, and leads to a concrete description of a large direct summand of $K_n(\mathbb{Z}[G])\otimes_{\mathbb{Z}}\mathbb{Q}$ 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 homological finiteness assumptions on the group only. The proof uses the cyclotomic trace map to topological cyclic homology, B\"okstedt-Hsiang-Madsen's functor C, and new general isomorphism and injectivity results about the assembly maps for topological Hochschild homology and C.Comment: To appear in Advances in Mathematics. 77 page

\'Etale homotopy equivalence of rational points on algebraic varieties

It is possible to talk about the \'etale homotopy equivalence of rational points on algebraic varieties by using a relative version of the \'etale homotopy type. We show that over $p$-adic fields rational points are homotopy equivalent in this sense if and only if they are \'etale-Brauer equivalent. We also show that over the real field rational points on projective varieties are \'etale homotopy equivalent if and only if they are in the same connected component. We also study this equivalence relation over number fields and prove that in this case it is finer than the other two equivalence relations for certain generalised Ch\^atelet surfaces.Comment: New title, rewritten introduction, 48 pages. To appear in Algebra & Number Theor

Li-Yorke Chaos for Composition Operators on LpL^p-Spaces

Li-Yorke chaos is a popular and well-studied notion of chaos. Several simple and useful characterizations of this notion of chaos in the setting of linear dynamics were obtained recently. In this note we show that even simpler and more useful characterizations of Li-Yorke chaos can be given in the special setting of composition operators on $L^p$ spaces. As a consequence we obtain a simple characterization of weighted shifts which are Li-Yorke chaotic. We give numerous examples to show that our results are sharp

Thick hyperbolic 3-manifolds with bounded rank

We construct a geometric decomposition for the convex core of a thick hyperbolic 3-manifold M with bounded rank. Corollaries include upper bounds in terms of rank and injectivity radius on the Heegaard genus of M and on the radius of any embedded ball in the convex core of M.Comment: 170 pages, 17 figure

The Cyclic and Epicyclic Sites

We determine the points of the epicyclic topos which plays a key role in the geometric encoding of cyclic homology and the lambda operations. We show that the category of points of the epicyclic topos is equivalent to projective geometry in characteristic one over algebraic extensions of the infinite semifield of max-plus integers. An object of this category is a pair of an algebraic extension of the semifield and an archimedean semimodule over this extension. The morphisms are projective classes of semilinear maps between semimodules. The epicyclic topos sits over the arithmetic topos which we recently introduced and the fibers of the associated geometric morphism correspond to the cyclic site. In two appendices we review the role of the cyclic and epicyclic toposes as the geometric structures supporting cyclic homology and the lambda operations.Comment: 35 pages, 5 figure