Kazhdan groups of dimension $16$ with prescribed second $\ell^2$-Betti number
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909024375537664 |
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| author | Fournier-Facio, Francesco Sauer, Roman |
| author_facet | Fournier-Facio, Francesco Sauer, Roman |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_00074 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Kazhdan groups of dimension $16$ with prescribed second $\ell^2$-Betti number Fournier-Facio, Francesco Sauer, Roman Group Theory 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. |
| title | Kazhdan groups of dimension $16$ with prescribed second $\ell^2$-Betti number |
| topic | Group Theory |
| url | https://arxiv.org/abs/2601.00074 |