On the algebraic \(K\)- and \(L\)-theory of word hyperbolic groups.
Let \(\Gamma\) be a discrete group, \(\underline E\Gamma\) be a universal space for proper actions for \(\Gamma\). The authors prove the following theorem: Theorem 1.1. Let \(\Gamma\) be a word hyperbolic group. Then (i) the assembly map \(H^\Gamma_*(\underline E\Gamma;\mathbb{K}^{-\infty}(R\Gamma_x))\to K_*(R\Gamma)\) in algebraic \(K\)-theory is a split injection for any ring with unit \(R\); (ii) the assembly map \(H^\Gamma_*(\underline E\Gamma;\mathbb{L}^{-\infty}(R\Gamma_x))\to L^{\langle-\infty\rangle}_*(R\Gamma)\), in algebraic \(L\)-theory, is a split injection for any ring with involution \(R\) such that for sufficiently large \(i\), \(K_{-i}(RH)=0\) for every finite subgroup \(H\) of \(\Gamma\). The proof is by showing that an appropriate compactification \(\overline X\) of the Rips complex, \(X\), associated to \(\Gamma\) satisfies the following properties: (i) the action of \(\Gamma\) extends to \(\overline X\); (ii) \(\overline X\) is metrizable; (iii) \(X^H\) is dense in \(\overline X^H\) for every finite subgroup \(H\) of \(\Gamma\); (iv) \(X^H\) is contractible for every finite subgroup \(H\) of \(\Gamma\); (v) compact subsets of \(X\) become small near \(\overline X-X\). The authors use that the above suffices by the results of \textit{D. Rosenthal} [K-Theory 32, No. 2, 139-166 (2004; Zbl 1068.19008) and Forum Math. 18, No. 2, 193-209 (2006; Zbl 1115.18006)].
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