Squeezing and higher algebraic \(K\)-theory (Q1398053)

Let \(R\) be an associative ring with unit and let \(\Gamma\) be a group of finite asymptotic dimension admitting a finite classifying space \(B\Gamma\). The author proves that the assembly map \[ H_*(B\Gamma;\mathbb K^{-\infty}R)\to K_*(R\Gamma) \] is split injective. Here \(\mathbb K^{-\infty}R\) is the nonconnective version of the K-theory spectrum of \textit{E. K. Pedersen} and \textit{C. A. Weibel} [Lect. Notes Math. 1126, 166--181 (1985; Zbl 0591.55002)]. This is an algebraic K-theory analog of a theorem of \textit{G. Yu} [Ann. Math. (2) 147, 325--355 (1998; Zbl 0911.19001)]. The proof uses controlled K-theory. The paper concludes with a brief discussion of L-theory.