I am currently reading Ghys and Sergiescu's paper [Sur un groupe remarquable de difféomorphisms du cercle](http://perso.ens-lyon.fr/ghys/articles/grouperemarquable.pdf) (French only I'm afraid), but part of their proof of Corollary 3.4 (page 20 of the pdf) is opaque to me.


Some notation first though. Let $K = \operatorname{GA}(\Bbb Q\_2)$, the general affine group of the dyadic rationals, and $K\_0$ the stabilizer of $0$ with respect to the natural action of $K$ on $\Bbb R$. Let $\Gamma$ denote the pseudogroup on $\Bbb R$ of all local homeomorphisms that are piecewise $K$. Then $\operatorname B\Gamma$ is the classifying space of $\Gamma$.


Ghys and Sergiescu claim that $\pi\_1(\mathrm{B}\Gamma) = 0$ and that, to prove this, it suffices to show that the normalizer of $K\_0$ is equal to $K$. Why is that sufficient?



A sufficient condition for $\pi_1(\operatorname B\Gamma) = 0$
Peter Greenberg's paper [Pseudogroups from group actions](https://www.jstor.org/stable/2374493) held the key.


There is a weak equivalence between $\mathrm{B}\Gamma$ and $(\mathrm{B}K\_0 \ast \mathrm{B}K\_0) \vee (\mathrm{B}K/\mathrm{B}K\_0)$, where $\ast$ is the join, $\vee$ is the usual wedge sum, and $\mathrm{B}K/\mathrm{B}K\_0$ denotes the homotopy pushout of the diagram
$$
\mathrm{C}\mathrm{B}K\_0 \leftarrow \mathrm{B}K\_0 \rightarrow \mathrm{B}K,
$$
where $\mathrm{C}$ denotes the cone of a space.


As $K\_0 \cong \Bbb Z$ this becomes $\mathrm{B}\Gamma \simeq S^3 \vee (\mathrm{B}K/\mathrm{B}K\_0)$, so one need only compute the fundamental group of $\mathrm{B}K/\mathrm{B}K\_0$. A quick invocation of Seifert-van Kampen reveals that
$$
\pi\_1(\mathrm{B}K/\mathrm{B}K\_0) \cong \pi\_1(\mathrm{B}K)/N(\pi\_1(\mathrm{B}K\_0)) \cong K/N(K\_0).
$$


