Cost of one-relator groups

Antoine Poulin; Konrad Wróbel

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arXiv · 2606.23506

Cost of one-relator groups

Antoine Poulin·Konrad Wróbel

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Abstract

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

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BibTeX RIS

Antoine Poulin, Konrad Wróbel. 2026-06-22. Cost of one-relator groups. https://arxiv.org/abs/2606.23506

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