$\ell^2$ Betti numbers and coherence of random groups

Dawid Kielak; Robert Kropholler; Gareth Wilkes

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arXiv · 2003.06354

$\ell^2$ Betti numbers and coherence of random groups

Dawid Kielak·Robert Kropholler·Gareth Wilkes

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Abstract

We study $\ell^2$ Betti numbers, coherence, and virtual fibring of random groups in the few-relator model. In particular, random groups with negative Euler characteristic are coherent, have $\ell^2$ homology concentrated in dimension 1, and embed in a virtually free-by-cyclic group with high probability. Similar results are shown with positive probability in the zero Euler characteristic case.

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Dawid Kielak, Robert Kropholler, Gareth Wilkes. 2020-03-13. $\ell^2$ Betti numbers and coherence of random groups. https://arxiv.org/abs/2003.06354

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