Cost of one-relator groups — Draconitic Research

Abstract

For any infinite one-relator group Γ=⟨ S | w^m⟩, we prove that cost(Γ)=|S|−1/m. For such groups, this gives β⁽²⁾₁(Γ)=cost(Γ) − 1, answering a special case of Gaboriau's question on the relationship between cost and first ℓ²-Betti number.

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BibTeX RIS

BibTeX

@misc{poulin2026cost,
  title = {{Cost of one-relator groups}},
  author = {Antoine Poulin and Konrad Wróbel},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2606.23506},
  eprint = {2606.23506},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.23506},
}

RIS

TY  - UNPB
TI  - Cost of one-relator groups
AU  - Antoine Poulin
AU  - Konrad Wróbel
PY  - 2026
DA  - 2026/11/02
PB  - arXiv
DO  - 10.48550/arxiv.2606.23506
UR  - https://arxiv.org/abs/2606.23506
AB  - For any infinite one-relator group Γ=⟨ S | w^m⟩, we prove that cost(Γ)=|S|−1/m. For such groups, this gives β^(2)_1(Γ)=cost(Γ) − 1, answering a special case of Gaboriau's question on the relationship between cost and first ℓ^2-Betti number.
ER  -

Abstract

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