Expanders and box spaces

Abstract

We consider box spaces of finitely generated, residually finite groups $G$, and try to distinguish them up to coarse equivalence. We show that, for $n\ge 2$, the group $S{L}_n(\mathbb{Z})$ has a continuum of box spaces which are pairwise non-coarsely equivalent expanders. Moreover, varying the integer $n\ge 3$, expanders given as box spaces of $S{L}_n(\mathbb{Z})$ are pairwise inequivalent; similarly, varying the prime $p$, expanders given as box spaces of $S{L}_2(\mathbb{Z}[\sqrt{p}])$ 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 ${\mathbb{Z}}^2⋊\mathbb{Z}$, 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.

23 pages; comments welcome

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