of the group $G$ [@petersen-sauer-thom]. Recently Carderi, Gaboriau and de la Salle proved a theorem for graphed groupoids and ultra products of them, which for instance implies the convergence for Farber sequences [@carderi].

random rooted simplicial complexes {#rrsc}
==================================

Let ${\mathcal{SC}_*}$ be the space of isomorphism classes of connected locally finite rooted simplicial complexes. We denote an element of ${\mathcal{SC}_*}$ with representative $(K,x)$ by $[K,x]$. We write $B_{(K,x)}(r)$ for the closed ball of radius $r$ centred at $x$ in $K$, i.e. the rooted subcomplex of $K$ consisting of all simplices of $K$ such that the graph distance of all their vertices to the root is at most $r$. We denote the subset of ${\mathcal{SC}_*}$ of all possible rooted isomorphism classes of $r$-balls by $\mathcal{B}(r)$. A metric $d$ on ${\mathcal{SC}_*}$ is given by $$d([K,x],[L,y])=\inf_{r}\left\lbrace \frac{1}{2^r}\;\middle|\;B_{(K,x)}(r)\cong B_{(L,y)}(r)\right\rbrace$$ and we define the $r$-neighbourhood of $[K,x]$ by $$\begin{aligned}
U_r(K,x):&=\lbrace[L,y]\in{\mathcal{SC}_*}\;|\; B_{(L,y)}(r)\cong B_{(K,x)}(r)\rbrace \\
&=\Big\lbrace[L,y]\in{\mathcal{SC}_*}\;|\; d([L,y],[K,x])\le \frac{1}{2^r}\Big\rbrace.\end{aligned}$$ With the $r$-neighbourhoods of finite simplicial complexes as basis, ${\mathcal{SC}_*}$ becomes a Polish space. The isomorphism classes of finite rooted simplicial complexes are a dense and countable subset of ${\mathcal{SC}_*}$. Moreover, the subset ${\mathcal{SC}_*}^d$ of isomorphism classes of rooted simplicial complexes with vertex degree bounded by $d\in{{\mathbb N}}$ is 