Fundamental groups of asymptotic cones of Lie groups with the SOL obstruction

Abstract

We study asymptotic cones of Lie groups, presenting a link between the geometry of the weights and the highly non-simply-connected nature of asymptotic cones. We show that a Lie group that admits a group of SOL-type as a quotient is such that the fundamental group of its asymptotic cone contains the fundamental group of the Hawaiian earring space. We also obtain a strong non-vanishing result for the first homology group of the asymptotic cone. The assumption on the Lie group, introduced by Abels and called the "SOL obstruction", is equivalent to an explicit geometric condition on the weights. Our work builds upon the results of Burillo, who proves the statement on asymptotic cones in the case of SOL, and on those of Cornulier and Tessera who show that the SOL obstruction implies exponential growth of the Dehn function.

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BibTeX RIS

BibTeX

@misc{velut2026fundamental,
  title = {{Fundamental groups of asymptotic cones of Lie groups with the SOL obstruction}},
  author = {Antoine Velut},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.30767},
  eprint = {2608.30767},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.30767},
}

RIS

TY  - UNPB
TI  - Fundamental groups of asymptotic cones of Lie groups with the SOL obstruction
AU  - Antoine Velut
PY  - 2026
DA  - 2026/08/31
PB  - arXiv
DO  - 10.48550/arxiv.2608.30767
UR  - https://arxiv.org/abs/2608.30767
AB  - We study asymptotic cones of Lie groups, presenting a link between the geometry of the weights and the highly non-simply-connected nature of asymptotic cones. We show that a Lie group that admits a group of SOL-type as a quotient is such that the fundamental group of its asymptotic cone contains the fundamental group of the Hawaiian earring space. We also obtain a strong non-vanishing result for the first homology group of the asymptotic cone. The assumption on the Lie group, introduced by Abels and called the "SOL obstruction", is equivalent to an explicit geometric condition on the weights. Our work builds upon the results of Burillo, who proves the statement on asymptotic cones in the case of SOL, and on those of Cornulier and Tessera who show that the SOL obstruction implies exponential growth of the Dehn function.
ER  -

Abstract

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