Expanders and box spaces

We consider box spaces of finitely generated, residually finite groups $G$, and try to distinguish them up to coarse equivalence. We show that, for $n\geq 2$, the group $SL_n(\mathbb{Z})$ has a continuum of box spaces which are pairwise non-coarsely equivalent expanders. Moreover, varying the integer $n\geq 3$, expanders given as box spaces of $SL_n(\mathbb{Z})$ are pairwise inequivalent; similarly, varying the prime $p$, expanders given as box spaces of $SL_2(\mathbb{Z}[\sqrt{p}])$ are pairwise inequivalent. A strong form of non-expansion for a box space is the existence of $\alpha\in]0,1]$ such that the diameter of each component $X_n$ satisfies $diam(X_n)=\Omega(|X_n|^\alpha)$. By a result of Breuillard and Tointon, the existence of such a box space implies that $G$ virtually maps onto $\mathbb{Z}$: we establish the converse. For the lamplighter group $(\mathbb{Z}/2\mathbb{Z})\wr\mathbb{Z}$ and for a semi-direct product $\mathbb{Z}^2\rtimes\mathbb{Z}$, such box spaces are explicitly constructed using specific congruence subgroups. We finally introduce the full box space of $G$, i.e. the coarse disjoint union of all finite quotients of $G$. We prove that the full box space of a group mapping onto the free group $\mathbb{F}_2$ is not coarsely equivalent to the full box space of an $S$-arithmetic group satisfying the Congruence Subgroup Property.