I am trying to understand proposition II.3.1 of http://arxiv.org/pdf/1309.2034v6.pdf, but I have some difficulties.
I get the construction of the permutations $\sigma_{\gamma}$ for each $\gamma\in F$. It follows that there is a well-defined map $\phi:F\rightarrow S_{|K|}:\gamma\mapsto \sigma_{\gamma}$. We want to show that this map is an $(F,2\varepsilon)$-approximate homomorphism. Now first of all we need to extend $\phi$ to all of $\Gamma$, I am not sure how to do this. But more importantly, we need to show that $d_{S_{\left|K\right|}}(\phi(g),Id)\geq 1-2\varepsilon$. I find the following:
$d_{S_{\left|K\right|}}(\phi(g),Id)= \frac{\left|\left\{x\in K\mid \sigma_g(x)\neq x\right\}\right|}{\left|K\right|}\geq \frac{\left|\left\{x\in g^{-1}K\cap K\mid gx\neq x\right\}\right|}{\left|K\right|}=\frac{\left|g^{-1}K\cap K\right|}{\left|K\right|}>\frac{\left|gK\cup K\right|}{\left|K\right|}-\varepsilon$.
Here I used the Folner condition in the last inequality. I am puzzled, this doesn't look right to me.
Similarly we need to show that $d_{S_{\left|K\right|}}(\phi(g)\phi(h),\phi(gh))<2\varepsilon$ for all $g,h\in F$. This already fails since I don't know whether $\phi(gh)$ is defined.
Any clearification would be nice. I've been stuck on showing that amenable groups are sofic for far too long.
Thanks