The integral K-theoretic Novikov conjecture for groups with finite asymptotic dimension

Abstract

Abstract. The integral assembly map in algebraic K-theory is split injective for any geometrically finite discrete group with finite asymptotic dimension.The goal of this paper is to apply the techniques developed by the first author in [3] to verify the integral Novikov conjecture for groups with finite asymptotic dimension as defined by M. Gromov [9].Recall that a finitely generated group Γ can be viewed as a metric space with the word metric associated to a given presentation. Definition (Gromov). A family of subsets in a general metric spaceThe asymptotic dimension of X is defined as the smallest number n such that for any d > 0 there is a uniformly bounded cover U of X by n + 1 d-disjoint families of subsets U = U 0 ∪ . . . ∪ U n .

On the K-theory of groups with finite asymptotic dimension

“…In [Bar03b], the first author proved the algebraic K-and L-theory versions of Yu's work by developing a squeezing theorem for higher algebraic K-theory to use with the approach established in [Yu98]. For algebraic K-theory, this was also achieved in [CG04a].…”

“…This work was supported by the SFB 478 "Geometrische Strukturen in der Mathematik". considered in [Bar03b,CG04a], factors through the assembly map (0.1) via (0.3) H n (BΓ; K R ) ∼ = H Γ n (EΓ; K R ) → H Γ n (EΓ; K R ). In particular, the cokernel of (0.2) contains the cokernel of (0.3) in the situation of Theorem A.…”

Integral Novikov conjectures and arithmetic groups containing torsion elements

“…When Γ is torsion free, it is reduced to the result that asdim Γ < +∞ and the existence of a finite BΓ imply the integral Novikov conjectures for Γ, proved in [6,7,8,35,36].…”

A notion of geometric complexity and its application to topological rigidity

“…2, is inspired by the property of finite asymptotic dimension of Gromov [18], of which it is a far reaching generalization [20]. The concept of finite asymptotic dimension has played an important role in the computation of K-theory in both topological and analytic contexts [2,5,6,28].…”