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

Francesco Fournier-Facio; Roman Sauer

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arXiv · 2601.00074

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

Francesco Fournier-Facio·Roman Sauer

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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.

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Francesco Fournier-Facio, Roman Sauer. 2025-12-31. Kazhdan groups of dimension $16$ with prescribed second $\ell^2$-Betti number. https://arxiv.org/abs/2601.00074

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