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

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