Kazhdan groups of dimension $16$ with prescribed second $\ell^2$-Betti number
Abstract
We construct a family of simple, lacunary hyperbolic groups with property that have rational cohomological dimension~ and whose second -Betti number can be prescribed to be any positive real. Moreover, we construct hyperbolic groups with property whose second -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 -Betti number is far from being semi-continuous in the space of marked groups, even assuming good finiteness properties.
Keywords
Cite
@article{arxiv.2601.00074,
title = {Kazhdan groups of dimension $16$ with prescribed second $\ell^2$-Betti number},
author = {Francesco Fournier-Facio and Roman Sauer},
journal= {arXiv preprint arXiv:2601.00074},
year = {2026}
}
Comments
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