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arXiv · 2610.29502

The first $\ell^2$-Betti number, cost and rank gradient of few-relator random groups

Miguel Donoso-Echenique·Eduardo Silva
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Abstract

Let $G$ be a random group in the few-relator model with $n\ge 2$ generators and $m\ge 1$ relators. We prove that asymptotically almost surely the first $\ell^2$-Betti number, the cost minus one, and the absolute rank gradient of $G$ are all equal to $\max\{n-m-1,0\}$. In particular, the equality $β_1^{(2)}(G)=0$ when $m\ge n-1$ answers positively a conjecture of Kielak, Kropholler and Wilkes [J. Lond. Math. Soc., 2022].

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Miguel Donoso-Echenique, Eduardo Silva. 2026-10-21. The first $\ell^2$-Betti number, cost and rank gradient of few-relator random groups. https://arxiv.org/abs/2610.29502

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