Free loop space of classifying space as a disjoint union of classifying spaces of centralizer proof reference request.

Question

I am looking for a reference for the proof or explanation of why for a discrete group $G$ we have that the free loop space of its classifying space is the disjoint union of centralizeers of $g$ where $[g]$ is the conjugacy class of elements in $G$, or more abbreviated,

$L(BG)= \bigsqcup_{[g]}BC_g$.

Answer 1

This answer on mathoverflow by Craig Westerland gives a proof of the result you want. It uses the following result: $$L(BG) \simeq G^{ad} \times_G EG$$ for which references are given in this answer by Dan Ramras.