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PreprintCC BY 4.0arXiv

ℓ2-Betti numbers and Σ-invariants of graph products of cyclic groups and their automorphisms

  • Marcos Escartín-Ferrer
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Abstract

We study the ℓ2-Betti numbers of graph products of cyclic groups. We relate these invariants to the ℓ2-Betti numbers of right-angled Artin groups and obtain explicit formulas in several cases determined by the structure of the finite-order vertices in the defining graph. In particular, we consider the cases where these vertices form a clique or have identical links, and derive a formula for the first ℓ2-Betti number when the defining graph has exactly two finite-order vertices. We also compute the BNSR-invariants of graph products of cyclic groups and the first ℓ2-Betti number of their automorphism groups when the defining graph has at most two finite-order vertices.

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Cite

BibTeXRIS
BibTeX
@misc{escartinferrer2026ℓ2betti,
  title = {{$\ell^2$-Betti numbers and $Σ$-invariants of graph products of cyclic groups and their automorphisms}},
  author = {Marcos Escartín-Ferrer},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.09704},
  eprint = {2610.09704},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.09704},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - ℓ^2-Betti numbers and Σ-invariants of graph products of cyclic groups and their automorphisms
AU  - Marcos Escartín-Ferrer
PY  - 2026
DA  - 2026/10/07
PB  - arXiv
DO  - 10.48550/arxiv.2610.09704
UR  - https://arxiv.org/abs/2610.09704
AB  - We study the ℓ^2-Betti numbers of graph products of cyclic groups. We relate these invariants to the ℓ^2-Betti numbers of right-angled Artin groups and obtain explicit formulas in several cases determined by the structure of the finite-order vertices in the defining graph. In particular, we consider the cases where these vertices form a clique or have identical links, and derive a formula for the first ℓ^2-Betti number when the defining graph has exactly two finite-order vertices. We also compute the BNSR-invariants of graph products of cyclic groups and the first ℓ^2-Betti number of their automorphism groups when the defining graph has at most two finite-order vertices.
ER  -