Seminar

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).