Expanders and box spaces

TL;DR

This paper studies box spaces of finitely generated residually finite groups, showing how they can be distinguished up to coarse equivalence, and explores their properties related to expansion, group structure, and full box spaces.

Contribution

It constructs explicit examples of non-coarsely equivalent expanders from $SL_n(bZ)$ and relates box space properties to group mappings and algebraic properties.

Findings

Multiple non-coarsely equivalent expanders from $SL_n(bZ)$ for $n eq 2$

Characterization of box spaces with strong non-expansion properties

Distinction between full box spaces of groups mapping onto free groups and $S$-arithmetic groups

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 $n\geq 2$, 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.