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PreprintCC BY 4.0arXiv

Commensurability and quasi-isometry classification for one vertex one loop tubular groups

  • Amy Tao
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Abstract

A tubular group has a graph of groups decomposition with ℤ2 vertex groups and ℤ edge groups. This paper gives a classification of one vertex one loop tubular groups G(k,ℓ),(m,n) = ⟨ a,b,t: [a,b]=1, t am bn t−1 = ak bℓ ⟩ up to commensurability and quasi-isometry. We show that tubular groups where the images of the edge maps have nonzero ``intersection number" are all commensurable and so quasi-isometric. When the intersection number is zero, there are two quasi-isometry classes and infinitely many commensurability classes. The nonzero and zero intersection number tubular groups are not quasi-isometric.

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Cite

BibTeXRIS
BibTeX
@misc{tao2026commensurability,
  title = {{Commensurability and quasi-isometry classification for one vertex one loop tubular groups}},
  author = {Amy Tao},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.05705},
  eprint = {2609.05705},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.05705},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Commensurability and quasi-isometry classification for one vertex one loop tubular groups
AU  - Amy Tao
PY  - 2026
DA  - 2026/09/04
PB  - arXiv
DO  - 10.48550/arxiv.2609.05705
UR  - https://arxiv.org/abs/2609.05705
AB  - A tubular group has a graph of groups decomposition with ℤ^2 vertex groups and ℤ edge groups. This paper gives a classification of one vertex one loop tubular groups G_(k,ℓ),(m,n) = ⟨ a,b,t: [a,b]=1, t a^m b^n t^−1 = a^k b^ℓ ⟩ up to commensurability and quasi-isometry. We show that tubular groups where the images of the edge maps have nonzero ``intersection number" are all commensurable and so quasi-isometric. When the intersection number is zero, there are two quasi-isometry classes and infinitely many commensurability classes. The nonzero and zero intersection number tubular groups are not quasi-isometric.
ER  -