A continuum of non-measure equivalent groups

We construct a continuum sized family $\{G_x{\}}_{x\in \{0,1{\}}^{\mathbb{N}}}$ of pairwise non-measure equivalent countable groups which have property (T) (hence are finitely generated), have zero ${\mathrm{\ell }}^2$-Betti numbers of all orders, and are torsion-free.