Fundamental Groups in the Five Dimensional Pan-Rong Conjecture

Abstract

Let M be a complete open $5$-manifold with nonnegative Ricci curvature. If its universal cover M̃ has Euclidean volume growth, then π₁(M) is finitely generated and virtually ℤ^k for some 0≤ k≤4. This confirms a conjecture of Pan and Rong \cite{PR18} in dimension five; see also \cite{BNS25,BrNaSe25}. In dimension six, under the same volume growth assumption, we also obtain a quantitative virtual abelianness result without assuming finite generation of the fundamental group.

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BibTeX RIS

BibTeX

@misc{huang2026fundamental,
  title = {{Fundamental Groups in the Five Dimensional Pan-Rong Conjecture}},
  author = {Hongzhi Huang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.09080},
  eprint = {2609.09080},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.09080},
}

RIS

TY  - UNPB
TI  - Fundamental Groups in the Five Dimensional Pan-Rong Conjecture
AU  - Hongzhi Huang
PY  - 2026
DA  - 2026/09/08
PB  - arXiv
DO  - 10.48550/arxiv.2609.09080
UR  - https://arxiv.org/abs/2609.09080
AB  - Let M be a complete open $5$-manifold with nonnegative Ricci curvature. If its universal cover M̃ has Euclidean volume growth, then π_1(M) is finitely generated and virtually ℤ^k for some 0≤ k≤4. This confirms a conjecture of Pan and Rong \cite{PR18} in dimension five; see also \cite{BNS25,BrNaSe25}. In dimension six, under the same volume growth assumption, we also obtain a quantitative virtual abelianness result without assuming finite generation of the fundamental group.
ER  -

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