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Quasi-isometries between graphs of totally disconnected locally compact groups

  • Sebastian Giersbach
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Abstract

Let G and H be compactly generated totally disconnected locally compact (tdlc) groups that decompose as finite graphs of tdlc groups (𝒢, 𝒜) and (ℋ, ℬ) such that all edge groups are compact and all vertex groups have at most one end. We generalize a result of Papasoglu--Whyte to tdlc groups and show that G and H are quasi-isometric if and only if they have the same number of ends and every one-ended vertex group of (𝒢, 𝒜) is quasi-isometric to a one-ended vertex group of (ℋ, ℬ), and vice versa. As an application, we construct uncountably many pairwise non-quasi-isometric compactly generated non-discrete simple tdlc groups, strengthening a result by Smith.

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Cite

BibTeXRIS
BibTeX
@misc{giersbach2026quasiisometries,
  title = {{Quasi-isometries between graphs of totally disconnected locally compact groups}},
  author = {Sebastian Giersbach},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.30881},
  eprint = {2608.30881},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.30881},
}
RIS
TY  - UNPB
TI  - Quasi-isometries between graphs of totally disconnected locally compact groups
AU  - Sebastian Giersbach
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2608.30881
UR  - https://arxiv.org/abs/2608.30881
AB  - Let G and H be compactly generated totally disconnected locally compact (tdlc) groups that decompose as finite graphs of tdlc groups (𝒢, 𝒜) and (ℋ, ℬ) such that all edge groups are compact and all vertex groups have at most one end. We generalize a result of Papasoglu--Whyte to tdlc groups and show that G and H are quasi-isometric if and only if they have the same number of ends and every one-ended vertex group of (𝒢, 𝒜) is quasi-isometric to a one-ended vertex group of (ℋ, ℬ), and vice versa. As an application, we construct uncountably many pairwise non-quasi-isometric compactly generated non-discrete simple tdlc groups, strengthening a result by Smith.
ER  -