Cost of one-relator groups

Abstract

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

arXiv comment12 pages