Expanders, especially those coming from box spaces of residually finite groups,
have been used to test various forms of the coarse Baum-Connes conjecture.
The first construction of a pair of expanders, one not coarsely embedding in the other,
was provided by Mendel and Naor in 2012. This was extended by Hume in 2014 who constructed a continuum
of expanders with unbounded girth, pairwise not coarsely equivalent. In joint work with A. Khukhro,
we construct a continuum of expanders with geometric property (T) of Willett-Yu, as box spaces of SL(3,Z).
We will discuss the following results: if box spaces of groups G, H are coarsely equivalent,
then the groups G, H are quasi-isometric (Khukhro and myself), and moreover G and H are uniformly
measure equivalent (K. Das).