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

Source 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].