The integral \(K\)-theoretic Novikov conjecture for groups with finite asymptotic dimension
This selfcontained paper is devoted to provide a proof of a result closely related to the integral \(K\)-theoretic Novikov conjecture for the case of groups with finite asymptotic dimension, using techniques developed by \textit{G. Carlsson} [``Bounded \(K\)-theory and the assembly map in algebraic \(K\)-theory'', in: Novikov conjectures, index theory and rigidity. Vol. 2, Lond. Math. Soc. Lect. Note Ser. 227, 5--127 (1995; Zbl 0957.19001)]. As defined by \textit{M. Gromov} [``Asymptotic invariants of finite groups'', in: Geometric group theory. Vol. 2, Lond. Math. Soc. Lect. Note Ser. 182 (1993; Zbl 0841.20039)], a pair of subspaces of a given metric space \(X\) are said to be \(d\)-disjoint if the distance between then is larger than \(d\). The asymptotic dimension \(X\) is defined to be the smallest number \(n\) such that there is a uniformly bounded cover \({\mathcal U}\) of \(X\) by \(n+1\) \(d\)-disjoint families of subsets \(U=U_1,\ldots,U_n\). Given a discrete group \(\Gamma\) and a ring \(R\), an element \(\gamma\in\Gamma\) may be interpreted as an isomorphism of the trivial \(R[\Gamma]\)-module, and to each isomorphism \(f\) of finitely generated \(R[\Gamma]\)-modules corresponds an isomorphism \(\gamma\otimes f\) of finitely generated \(R[\Gamma]\)-modules. From the main result in this paper it follows that this is a functor which induces an assembly map \(\alpha:B\Gamma_+\wedge K(R)\rightarrow K(R[\Gamma])\). Here \(B\Gamma\) is the compact universal space of the group \(\Gamma\), the map \(\alpha:B\Gamma_+\wedge K(R)\) is the nonconnective \(K\)-theory spectrum of the group \(\Gamma\) with coefficients in the nonconnective \(K\)-theory spectrum of \(R\) and \(K(R[\Gamma])\) is the nonconnective \(K\)-theory spectrum of the group ring \(R[\Gamma]\). The integral Novikov conjecture predicts that this map is a split injection for any \(R\) and any group \(\Gamma\) with \(B\Gamma\) being a finite \(CW\)-complex. The main result is the following theorem: Let \(\Gamma\) be a geometrically finite group with finite asymptotic dimension and let \(R\) be an arbitrary ring. Then the assembly map \(\alpha:h\left(\Gamma, K(R)\right)\rightarrow K\left(R[\Gamma]\right)\) from the homology of the group \(\Gamma\) with coefficients in the \(K\)-theory spectrum \(K(R)\) to the \(K\)-theory of the group ring \(R[\Gamma]\) is a split injection. The proof rests on a result by \textit{A. Dranishikov} [``On hypersphericity of manifolds with finite asymptotic dimension'', Trans. Am. Math. Soc. 355, 155--167 (2003; Zbl 1020.53025)] which asserts that the asymptotic dimension of a metric space \(M\) is finite if and only if there is a uniform embedding of \(M\) in a finite product of locally finite simplicial trees.
scientific article; zbMATH DE number 2130987