A localization algebra approach to the Baum-Connes conjecture for extensions

For an extension $1\rightarrow N \rightarrow Γ\xrightarrow{q} Γ/ N \rightarrow 1$ of discrete countable groups, it is known that the Baum-Connes conjecture with coefficients holds for $Γ$ if it holds for $Γ/ N$ and $q^{-1}(F)$ for any finite subgroup $F$ of $Γ/ N$. In this paper, we employ a localization algebra approach to prove this result.