domain $F=\bigsqcup_{j\in J} F_j$ of $\Sigma^{(0)}$ such that $\pi\colon F_j\to X$ is injective and $\sum_{j\in J} \pi(\nu(F_j))<\infty$. We denote the vertex over $x$ in $F_j$ by $f_j(x)$. A fundamental domain for the ordered simplices of $\Sigma^{(p)}$ is given by $$\begin{aligned}
    \{ (f,v_1,...,v_p)\in \Sigma^{(p)}\;|\; f\in F,\; v_i\in \Sigma^{(0)}\text{ for }1\le i\le p\}.\end{aligned}$$ Among these ordered simplices we can choose a representative for each unordered simplex, this gives us a fundamental domain $F'=\bigsqcup_{i\in I} F_i'$ of $\Sigma^{(p)}$ for the action of $\mathcal{R}\times \mathfrak{S}_p$ such that the projection $\pi\colon F'_i\to X$ is injective for each $i\in I$. The characteristic functions of these sets define a total sequence $\sigma_i$ for the direct integral $\int_X^\oplus C^{(2)}_p(\Sigma_x)d\nu$ and hence define a trace [@gaboriau p.120]. We compute the $\ell^2$-Betti numbers: $$\begin{aligned}
    &\beta^{(2)}_p(\mu_{\Sigma,\mathcal{R}})\\
    =&\int_{{\mathcal{SC}_*}}\sum_{\tau \in \mathcal{T}_p}\frac{\langle P \tau,\tau\rangle}{p+1}d\mu_{\Sigma,\mathcal{R}}([K,x]) \\
    =&\int_X \frac{1}{\sum_{j\in J} \nu(\pi(F_j))}\sum_{j\in J}\sum_{\tau \in \mathcal{T}_p}\chi_{\pi(F_j)}\frac{\langle P \tau,\tau\rangle([\Sigma_x,f_j(x)])}{p+1}d\nu\\
    =&\int_X \frac{1}{\sum_{j\in J} \nu(\pi(F_j))}\sum_{j\in J}\sum_{\substack{\sigma \in \Sigma_x^{(p)}\\ f_j(x)\in\sigma}}\chi_{\pi(F_j)}(x)\frac{\langle P \sigma,\sigma\rangle_{\Sigma_x}}{p+1}d\nu\\
    =&\frac{1}{\sum_{j\in J} \nu(\pi(F_j))}\int_X \chi_{\pi(F'_i)}(x) \sum_{i\in I}\langle P \sigma_i(x),\sigma_i(x)\rangle_{\Sigma_x}d\nu\\
    =&\frac{{\operatorname{dim}}_\mathcal{R}\mathcal{H}_p^{(2)}(\Sigma)}{\sum_{j\in J} \nu(\pi(F_j))}=\frac{\beta_p(\Sigma,\mathcal{R},\nu)}{\sum_{j\in J} \nu(\pi(F_j))}\end{aligned}$$ As in the examples before, we used the fact that summing over all $p$-simplices containing one of the vertices of the fundamental domain $F$, is the same as summing $(p+1)$ times over a fundamental domain $F'$ for 