Kazhdan groups of dimension $16$ with prescribed second ${\ell }^2$-Betti number

Abstract

We construct a family of simple, lacunary hyperbolic groups with property $(T)$ that have rational cohomological dimension~$16$ and whose second ${\ell }^2$-Betti number can be prescribed to be any positive real. Moreover, we construct hyperbolic groups with property $(T)$ whose second ${\ell }^2$-Betti number can be prescribed to be any non-negative rational. Along the way, we present new constructions of measurably diverse finitely generated groups, and we prove that the second ${\ell }^2$-Betti number is far from being semi-continuous in the space of marked groups, even assuming good finiteness properties.

25 pages. v2: improved, now the second l2-Betti numbers of the groups in Theorem A are not just in an uncountable range, but can be any positive real (hence the change in the title). Moreover the new Theorem B constructs hyperbolic groups with prescribed rational second l2-Betti number