[Submitted on 22 Jun 2026]

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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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From: Konrad Wróbel [view email]
[v1] Mon, 22 Jun 2026 15:54:49 UTC (20 KB)

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