On an uncountable family of simple Kazhdan groups in dimension 16

We construct a family of simple Kazhdan groups that have rational cohomological dimension 16 and uncountably many values of second l2-Betti numbers.
Along the way, we present new constructions of measurably diverse finitely generated groups, and we prove that the second l2-Betti number is far from being
semi-continuous in the  space of marked groups.
 
The construction relies on four ingredients:
the theory of group-theoretic Dehn fillings (Osin and many others)
the Cohen-Lyndon property and its excision principle (Petroysan-Sun)
higher property T (Bader-Sauer)
the algebraic approach to l2-Betti numbers (Lück).  

This is joint work with Francesco Fournier-Facio.