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

TL;DR

This paper constructs hyperbolic groups with property (T) that have a prescribed second  ext{-}Betti number, demonstrating new diversity and non-semicontinuity properties in the space of groups.

Contribution

It introduces methods to construct hyperbolic groups with property (T) with any positive or non-negative rational second  ext{-}Betti number, expanding understanding of their cohomological properties.

Findings

Constructed hyperbolic groups with property (T) and prescribed second  ext{-}Betti number.

Showed the second  ext{-}Betti number is not semi-continuous in the space of marked groups.

Presented new constructions of diverse finitely generated groups.

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.