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 SL n (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 SL n (Z) are pairwise inequivalent; similarly, varying the prime p, expanders given as box spaces of SL 2 (Z[ √ p]) are pairwise inequivalent.A strong form of non-expansion for a box space is the existence of α ∈]0, 1] such that the diameter of each component X n satisfies diam(X n ) = Ω(|X n | α ). By [BT15], 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 Z 2 ⋊ 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 F 2 is not coarsely equivalent to the full box space of an S-arithmetic group satisfying the Congruence Subgroup Property.Definition 2. Fix α ∈]0, 1]. We say that (M k ) G has property D α if there exists K > 0 (only depending on S) such that, for every component G/M of (M k ) G: diam(G/M) ≥ K.|G/M| α .

We construct box spaces of a free group that do not coarsely embed into a Hilbert space, but do not contain coarsely nor weakly embedded expanders. We do this by considering two sequences of subgroups of the free group: one which gives rise to a box space which forms an expander, and another which gives rise to a box space that can be coarsely embedded into a Hilbert space. We then take certain intersections of these subgroups, and prove that the corresponding box space contains generalized expanders. We show that there are no weakly embedded expanders in the box space corresponding to our chosen sequence by proving that a box space that covers another box space of the same group that is coarsely embeddable into a Hilbert space cannot contain weakly embedded expanders.

We investigate how coarse embeddability of box spaces into Hilbert space behaves under group extensions. In particular, we prove a result which implies that a semidirect product of a finitely generated free group by a finitely generated residually finite amenable group has a box space which coarsely embeds into Hilbert space. This provides a new class of examples of metric spaces with bounded geometry which coarsely embed into Hilbert space but do not have property A, generalising the example of Arzhantseva, Guentner and Spakula.

We use a coarse version of the fundamental group first introduced by Barcelo, Kramer, Laubenbacher and Weaver to show that box spaces of finitely presented groups detect the normal subgroups used to construct the box space, up to isomorphism. As a consequence we have that two finitely presented groups admit coarsely equivalent box spaces if and only if they are commensurable via normal subgroups. We also provide an example of two filtrations (Ni) and (Mi) of a free group F such that Mi > Ni for all i with [Mi : Ni] uniformly bounded, but with (N i ) F not coarsely equivalent to (M i ) F . Finally, we give some applications of the main theorem for rank gradient and the first ℓ 2 Betti number, and show that the main theorem can be used to construct infinitely many coarse equivalence classes of box spaces with various properties.

In this paper, we use geometric tools to study the structure of asymptotic expanders and show that a sequence of asymptotic expanders always admits a "uniform exhaustion by expanders". It follows that asymptotic expanders cannot be coarsely embedded into any L p -space, and that asymptotic expanders can be characterised in terms of their uniform Roe algebra. Moreover, we provide uncountably many new counterexamples to the coarse Baum-Connes conjecture. These appear to be the first counterexamples that are not directly constructed by means of spectral gaps. Finally, we show that vertex-transitive asymptotic expanders are actually expanders. In particular, this gives a C * -algebraic characterisation of expanders for vertex-transitive graphs.