Cost of one-relator groups

Abstract

For any infinite one-relator group $\hat{I}“=⟨S\mid w^m⟩$, we prove that $\mathrm{cost}(\hat{I}“)=|S|-\frac{1}{m}$. 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.

12 pages