add important reformulation of BC for groups

contents/coarse.tex

@@ -1008,7 +1008,7 @@ The basic result about the equivariant \K-homology of the the balanced product i
 
 The following theorem requires even longer computations:
 
-\begin{theorem}{\cite[Thm.~6.5.15]{WilletYu}}
+\begin{theorem}{\cite[Thm.~6.5.15]{WilletYu}}\label{thm:iso_equiv_X_to_X/G}
 	Sag $G$ acts freely on $X$.
 	Then there are $*$-homomorphisms $\Phi \colon \LocAlg(H_X)^G \to \LocAlg(H_{X/G})$ and $\Psi \colon \LocAlg(H_{X/G}) \to \LocAlg(H_X)^G$ which give mutually inverse $*$-isomorphisms on the quotients
 	\(
@@ -1163,6 +1163,15 @@ and we say, that $G$ satisfies the Baum--Connes conjecture, if this map is an is
 
 \todo[inline]{Should contain the following: Rips complexes, Classifying space variants for BCC for $G$, BCC for Euclidean space}
 
+\begin{theorem}
+	Assume $G$ is finitely generated and torsion free and admits a classifying space $\B G$ which is a finite CW complex.
+	Then the Baum--Connes assembly map for $G$ (acting on itself) identifies with the composition
+	\[
+		\K_*(\B G) \cong \K^G_*(\E G) \To{\mu_{\E G}} \K_* \enbrace*{\RoeAlg(\E G)^G} \cong \K_*(\Cstar_\rho(G))
+	\]
+	(where we used the isomorphisms from \cref{thm:iso_equiv_X_to_X/G,thm:equiv_roe_alg_group_cstar})
+\end{theorem}
+
 % section a_concrete_model_for_coarse_k_homology (end)
 
 \section{Applications} % (fold)

math.tex

@@ -290,6 +290,7 @@
 
 
 \newcommand{\B}[1]{B{#1}} % classifying spaces
+\newcommand{\E}[1]{E{#1}} % Universal cover
 
 \newcommand{\Inf}{I\mkern-.1mu n\mkern-.5mu f}
 \newcommand{\Infhat}{I\mkern-.1mu \widehat{n}\mkern-.5mu f}