Cost of one-relator groups

Abstract

For any infinite one-relator group $Γ=\langle S \mid w^m\rangle$, we prove that $\mathrm{cost}(Γ)=|S|-\frac{1}{m}$. For such groups, this gives $β^{(2)}_1(Γ)=\mathrm{cost}(Γ) - 1$, answering a special case of Gaboriau's question on the relationship between cost and first $\ell^2$-Betti number.