Twisted equivariant K-theory for proper actions of discrete groups

We will make a construction of twisted equivariant K-theory for proper actions of discrete groups by using ideas of Lück and Oliver (Topology 40:585–616, 2001) to expand a construction of Adem and Ruan (Comm. Math. Phys. 237:533–556, 2003).     

The large scale topology and algebraic K-theory of arithmetic groups

New compactifications of symmetric spaces of noncompact type X are constructed using the asymptotic geometry of the Borel–Serre enlargement. The controlled K-theory associated to these compactifications is used to prove the integral Novikov conjecture for arithmetic groups.     

Hopf Algebra Equivariant Cyclic Cohomology, K-theory and Index Formulas

For an algebra with an action of a Hopf algebra we establish the pairing between equivariant cyclic cohomology and equivariant K-theory for . We then extend this formalism to compact quantum group actions and show that equivariant cyclic cohomology is a target space for the equivariant Chern character of equivariant summable Fredholm modules. We prove an analogue of Julg's theorem relating equivariant K-theory to ordinary K-theory of the C^*-algebra crossed product, and characterize equivariant vector bundles on quantum homogeneous spaces.     

K-theory and intersection theory revisited

Another proof that the product structure on K-theory may be used to define the product structure on the Chow ring of a smooth variety over a field is presented. The virtue of this proof is that it is essentially a formal argument using natural properties of Quillen's spectral sequence, the K-theory product, cycle classes, and the classical intersection product.     

Operator norm localization for linear groups and its application to K-theory

We prove the operator norm localization property for linear groups. As an application we prove the coarse Novikov conjecture for box spaces of a linear group.     

K-theory of fine topological algebras, Chern character, and assembly

TheK-theory of the group algebra [] for a countable, discrete group is defined in terms of the simplicial ring of smooth simplices on [], where [] is given the fine topology with respect to its finite-dimensional, linear subspaces. The assembly map for this theory :K _* B K _* [] is studied and shown to be a rational injection. The proof uses the Connes-Karoubi Chern character fromK-theory of Banach algebras to cyclic homology, here generalized to any fine topological algebra, and proved to be multiplicative.     

Psl(2,q) quotients of some hyperbolic tetrahedral and coxeter groups

We classify quotients of type PSL(2,q) and PGL(2,q) with torsion-free kernel for four of the nine hyperbolic tetrahedral groups. Using this result, we give a classification of the quotients with torsion-free kernel of type PSL(2q) ×Z₂ of the associated Coxeter or reflection groups. These do not admit quotients of type PSL(2,q),PGL(2,q). We also study quotients of type PSL(2,q) and PGL(2,q) of the fundamental group of the hyperbolic 3-orbifold of minimal known volume.     

K-theory of reduced C*-algebras and rapidly decreasing functions on groups

We associate to any length function L on a group a space of rapidly decreasing functions on (in the l ² sense), denoted by H _L (). When H _L () is contained in the reduced C*-algebra C _r ^* () of (), then it is a dense *-subalgebra of C _r ^* () and we prove a theorem of A. Connes which asserts that under this hypothesis H _L () has the same K-theory as C _r ^* (). We introduce another space of rapidly decreasing functions on (in the l ¹ sense), denoted by H _L ^1, (), which is always a dense *-subalgebra of the Banach algebra l ¹(), and we show that H _L ^1, () has the same K-theory as l ¹().     

A non-Abelian K-theory and pseudo-isotopies of 3-manifolds

A non-Abelian version of algebraic K-theory, based on automorphism of free products rather than automorphisms of free modules, is considered and is related to pseudo-isotopies of 3-manifolds.Sometime after writing this paper I learned that some of the algebraic results in it were first proved by Gersten in [13]. Specifically each of Theorems 3.1, 3.2 and 5.1 in the special case when is the trivial group, and Theorem 3.3 and its corollaries are all results of Gersten. I have left the paper as it is for the sake of completeness and since the approach here often differs considerably from Gersten's.     

On the K-theory of ?G, G a non-Abelian group of order pq

This paper contains the construction of a generalized Mayer-Vietoris sequence for the rings in the title, which can be described as multiple pullback rings. A comparison between their K- and G-theories is given in the form of a long exact sequence.     

Nonabelian K-theory: The nilpotent class of K1 and general stability

A functorial filtration GL _n =S^–1L _n S⁰L _n S ⁱ L _n E _n of the general linear group GL _{n, n} 3, is defined and it is shown for any algebra A, which is a direct limit of module finite algebras, that S^–1 L _n (A)/S⁰L _n (A) is abelian, that S⁰L _n (A) S¹L _n (A) is a descending central series, and that S ⁱ L _n (A) = E _n(A) whenever i the Bass-Serre dimension of A. In particular, the K-functors k ₁ S ⁱ L _n =S ⁱ L _n /E _n are nilpotent for all i 0 over algebras of finite Bass-Serre dimension. Furthermore, without dimension assumptions, the canonical homomorphism S ⁱ L _n (A)/S ⁱ⁺¹ L _n (A)S ⁱ L _{n+ 1}(A)/S ⁱ⁺¹ L _{n + 1} (A) is injective whenever n i + 3, so that one has stability results without stability conditions, and if A is commutative then S⁰L _n (A) agrees with the special linear group SL _n (A), so that the functor S⁰L _n generalizes the functor SL _n to noncommutative rings. Applying the above to subgroups H of GL _n (A), which are normalized by E _n(A), one obtains that each is contained in a sandwich GL _n (A, ) H E _n(A, ) for a unique two-sided ideal of A and there is a descending S⁰L _n (A)-central series GL _n (A, ) S⁰L _n (A, ) S¹L _n (A, ) S ⁱ L _n (A, ) E _n(A, ) such that S ⁱ L _n (A, )=E _n(A, ) whenever i Bass-Serre dimension of A.Dedicated to Alexander Grothendieck on his sixtieth birthday     

Cohomology of affine Artin groups and applications

The result of this paper is the determination of the cohomology of Artin groups of type and with non-trivial local coefficients. The main result

is an explicit computation of the cohomology of the Artin group of type with coefficients over the module Here the first standard generators of the group act by -multiplication, while the last one acts by -multiplication. The proof uses some technical results from previous papers plus computations over a suitable spectral sequence. The remaining cases follow from an application of Shapiro's lemma, by considering some well-known inclusions: we obtain the rational cohomology of the Artin group of affine type as well as the cohomology of the classical braid group with coefficients in the -dimensional representation presented in Tong, Yang, and Ma (1996). The topological counterpart is the explicit construction of finite CW-complexes endowed with a free action of the Artin groups, which are known to be spaces in some cases (including finite type groups). Particularly simple formulas for the Euler-characteristic of these orbit spaces are derived.

     

Isovariant maps from free -manifolds to representation spheres

The isovariant version of Borsuk–Ulam type theorems has been studied by Wasserman and the first author. In this paper, first we consider the relation between the existence of C_n-isovariant maps from free C_n-manifolds to representation spheres and Borsuk–Ulam type inequalities for their dimensions. Our main result classifies the C_n-isovariant maps by C_n-isovariant homotopy types when a Borsuk–Ulam type inequality holds. For proving it, we use the multidegree of a C_n-equivariant map developed by the first author.     

Operator norm localization property of relative hyperbolic group and graph of groups

In this article we study the spaces which have operator norm localization property. We prove that a finitely generated group Γ which is strongly hyperbolic with respect to a collection of finitely generated subgroups {H₁,…,Hn} has operator norm localization property if and only if each Hi, i=1,2,…,n, has operator norm localization property. Furthermore we prove the following result. Let π be the fundamental group of a connected finite graph of groups with finitely generated vertex groups GP. If GP has operator norm localization property for all vertices P then π has operator norm localization property.     

Groups acting on buildings, operator K-theory, and Novikov''s conjecture

The paper is devoted to the study of the KK-theory of Bruhat-Tits buildings. We develop a theory which is analogous to the corresponding theory for manifolds of nonpositive sectional curvature. We construct a C ^*-algebra and a Dirac element associated to any simplicial complex. In the case of buildings, we construct, moreover, a dual Dirac element and compute its KK-products with the Dirac element. As a consequence, we prove the Novikov conjecture for discrete subgroups of linear adelic groups. In our study, we develop a KK-theoretic Poincaré duality for non-Hausdorff manifolds.Dedicated to Alexander Grothendieck on his sixtieth birthday     

Mixed Hodge structure in cyclic homology and algebraic K-theory, 1

Deligne defined the notion of a mixed Hodge structure (MHS) and proved that every quasiprojective variety over has a natural MHS on its cohomology. This paper establishes similar results for cyclic homology and the algebraic K-theory of simply connected quasi-projective varieties over . In the nonsimply connected case, an MHS is established on certain quotient groups of algebraic K-theory.Supported by a NSERC University Research Fellowship and operating grant.     

Algebraic K-Theory In Low Degree and The Novikov Assembly Map

We prove that the Novikov assembly map for a group factorizes,in ‘low homological degree’, through the algebraicK-theory of its integral group ring. In homological degree 2,this answers a question posed by N. Higson and P. Julg. As adirect application, we prove that if is torsion-free and satisfiesthe Baum-Connes conjecture, then the homology group H₁(; Z)injects in and in , for any ring A such that . If moreover B is of dimension lessthan or equal to 4, then we show that H₂(; Z) injects in and in , where A is as before, and ₂ is generated by the Steinberg symbols{,}, for . 2000 Mathematical Subject Classification: primary 19D55, 19Kxx,58J22; secondary: 19Cxx, 19D45, 43A20, 46L85.     

Groups acting on tree-graded spaces and splittings of relatively hyperbolic groups

Tree-graded spaces are generalizations of R-trees. They appear as asymptotic cones of groups (when the cones have cut-points). Since many questions about endomorphisms and automorphisms of groups, solving equations over groups, studying embeddings of a group into another group, etc. lead to actions of groups on the asymptotic cones, it is natural to consider actions of groups on tree-graded spaces. We develop a theory of such actions which generalizes the well-known theory of groups acting on R-trees. As applications of our theory, we describe, in particular, relatively hyperbolic groups with infinite groups of outer automorphisms, and co-Hopfian relatively hyperbolic groups.     

Factorization and symplectic uniton numbers for harmonic maps into symplectic groups

It is proved that any harmonic map ϕ : Ω →Sp(N) from a simply connected domain Ω ⊆R ²⋃ | ∞ | into the symplectic groupSp(N) ⊂U(2N) with finite uniton number can be factorized into a product of a finite number of symplectic unitons. Based on this factorization, it is proved that the minimal symplectic uniton number of ϕ is not larger thanN, and the minimal uniton number of ϕ is not larger than 2N - 1. The latter has been shown in literature in a quite different way.