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The first ℓ2-Betti number, cost and rank gradient of few-relator random groups

  • Miguel Donoso-Echenique
  • Eduardo Silva
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Abstract

Let G be a random group in the few-relator model with n≥ 2 generators and m≥ 1 relators. We prove that asymptotically almost surely the first ℓ2-Betti number, the cost minus one, and the absolute rank gradient of G are all equal to max{n−m−1,0}. In particular, the equality β1(2)(G)=0 when m≥ n−1 answers positively a conjecture of Kielak, Kropholler and Wilkes [J. Lond. Math. Soc., 2022].

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Cite

BibTeXRIS
BibTeX
@misc{donosoechenique2026first,
  title = {{The first $\ell^2$-Betti number, cost and rank gradient of few-relator random groups}},
  author = {Miguel Donoso-Echenique and Eduardo Silva},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.29502},
  eprint = {2610.29502},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.29502},
}
RIS
TY  - UNPB
TI  - The first ℓ^2-Betti number, cost and rank gradient of few-relator random groups
AU  - Miguel Donoso-Echenique
AU  - Eduardo Silva
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2610.29502
UR  - https://arxiv.org/abs/2610.29502
AB  - Let G be a random group in the few-relator model with n≥ 2 generators and m≥ 1 relators. We prove that asymptotically almost surely the first ℓ^2-Betti number, the cost minus one, and the absolute rank gradient of G are all equal to max{n−m−1,0}. In particular, the equality β_1^(2)(G)=0 when m≥ n−1 answers positively a conjecture of Kielak, Kropholler and Wilkes [J. Lond. Math. Soc., 2022].
ER  -