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≥2, the group SLn(Z) has a continuum of box spaces which are pairwise non-coarsely equivalent expanders. Moreover, varying the integer n≥3, expanders given as box spaces of SLn(Z) are pairwise not coarsely equivalent; similarly, varying the prime p, expanders given as box spaces of SL2(Z[p]) are pairwise not coarsely equivalent. We also show that, for prime p, the family of finite groups (PSL2(Z/pnZ))n≥1 can be turned in infinitely many ways (up to coarse equivalence) into a 6-regular expander.A strong form of non-expansion for a box space is the existence of α∈]0,1] such that the diameter of each component Xn satisfies diam(Xn)=Ω(|Xn|α). By [2], the existence of such a box space implies that G virtually maps onto Z: we establish the converse. For the lamplighter group (Z/2Z)≀Z and for a semi-direct product Z2⋊Z, such box spaces are explicitly constructed using specific congruence subgroups.We finally introduce the full box space of G, i.e. the metric disjoint union of all finite quotients of G. We prove that the full box space of a group mapping onto the free group F2 is not coarsely equivalent to the full box space of an S-arithmetic group satisfying the Congruence Subgroup Property.