the kernel of the $p$-th Laplace operator. We compute the $\ell^2$-Betti numbers of $\mu_L$: $$\begin{aligned}
    \beta^{(2)}_p(\mu_L)&=\int_{{\mathcal{SC}_*}}\sum_{\tau\in\mathcal{T}_p}\frac{\langle P \tau,\tau\rangle}{p+1}d\mu_L([K,x])\\
    &=\sum_{\substack{[K,x]\in{\mathcal{SC}_*}\\ K\cong L}}\mu_L([K,x])\sum_{\substack{\sigma \in (K,x)_{{\mathbb N}}(p) \\ 0\in\sigma}}\frac{\langle P \sigma,\sigma\rangle}{p+1}\\
    &=\sum_{\substack{[K,x]\in {\mathcal{SC}_*}\\ K\cong L}}\frac{|\{y\in V(L)\;|\;(L,y)\in [K,x]\}|}{|V(L)|}\sum_{\substack{\sigma \in (K,x)_{{\mathbb N}}(p) \\ 0\in\sigma}}\frac{\langle P \sigma,\sigma\rangle}{p+1}\\
    &=\frac{1}{|V(L)|}\sum_{y\in V(L)}\sum_{\substack{\sigma \in L(p) \\ y\in\sigma}}\frac{\langle P \sigma,\sigma\rangle}{p+1}\\
    &=\frac{1}{|V(L)|}\sum_{\sigma \in L(p)}\langle P \sigma,\sigma\rangle=\frac{b_p(L)}{|V(L)|}=\beta_p^{(2)}(L,\{e\})\end{aligned}$$ From line three to four we used the fact that $(L,y)$ and $((K,x)_{{\mathbb N}},0)$ are root isomorphic, so $$\sum_{\substack{\sigma \in (K,x)_{{\mathbb N}}(p) \\ 0\in\sigma}}\frac{\langle P \sigma,\sigma\rangle}{p+1}=\sum_{\substack{\sigma \in L(p) \\ y\in\sigma}}\frac{\langle P \sigma,\sigma\rangle}{p+1}.$$ Also note, if we sum over all $p$-simplices containing a fixed vertex and then sum over all vertices, we hit every $p$-simplex exactly p+1 times. We will use this fact constantly.

2.  Let $L$ be an infinite connected locally finite simplicial complex with a free, admissible and cocompact action of a discrete group $\Gamma$ with fundamental domain ${\mathcal F}$. $$\begin{aligned}
    \beta^{(2)}_p(\mu_{L})&=\int_{{\mathcal{SC}_*}}\sum_{\tau\in\mathcal{T}_p}\frac{\langle P\tau,\tau\rangle}{p+1}d\mu_{L}([K,x]) \\
    &=\sum_{\substack{[K,x]\in{\mathcal{SC}_*}\\ K\cong L}}\mu_L([K,x])\sum_{\substack{\sigma \in (K,x)_{{\mathbb N}}(p) \\ 0\in\sigma}}\frac{\langle P\sigma,\sigma\rangle}{p+1}\\
    &=\frac{1}{|{\mathcal F}|}\sum_{x\in {\mathcal F}}\sum_{\substack{\sigma \in L(p)\\ x\in\sigma}}\frac{\langle P \sigma,\sigma\rangle}{p+1}\\
    &=\frac{1}{|{\mathcal F}|}\sum_{\sigma \in {\mathcal F}(p)}\langle P \sigma,\sigma\rangle=\frac{\beta^{(2)}_p(L,\Gamma)}{|{\mathcal F}|}\end{aligned}$$

3.  Given a probability measure preserving equivalence relation $\mathcal{R}\subset X\times X$ on a standard Borel space $(X,\nu)$. Let $\Sigma$ be a $\mathcal{R}$-simplicial complex with fundamental 