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One boundary, infinitely geometries

  • Jiming Ma
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Abstract

Let N be a 3-dimensional integral homology sphere. We prove the existence of a sequence of 4-dimensional topologically negatively curved manifolds Mk, k=1,2,…, satisfying the following: (1) all aspherical manifolds Mk have the same universal cover UN depending only on N; (2) all fundamental groups π1(Mk) are hyperbolic and share the same Gromov boundary XN depending only on N; (3) the groups π1(Mk) represent infinitely many distinct quasi-isometry classes. The last conclusion, which represents the new feature, shows that negatively curved geometries in dimension 4 exhibit substantially richer structure than those in dimension 3.

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Cite

BibTeXRIS
BibTeX
@misc{ma2026boundary,
  title = {{One boundary, infinitely geometries}},
  author = {Jiming Ma},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.15406},
  eprint = {2610.15406},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.15406},
}
RIS
TY  - UNPB
TI  - One boundary, infinitely geometries
AU  - Jiming Ma
PY  - 2026
DA  - 2026/10/11
PB  - arXiv
DO  - 10.48550/arxiv.2610.15406
UR  - https://arxiv.org/abs/2610.15406
AB  - Let N be a 3-dimensional integral homology sphere. We prove the existence of a sequence of 4-dimensional topologically negatively curved manifolds M_k, k=1,2,…, satisfying the following: (1) all aspherical manifolds M_k have the same universal cover U_N depending only on N; (2) all fundamental groups π_1(M_k) are hyperbolic and share the same Gromov boundary X_N depending only on N; (3) the groups π_1(M_k) represent infinitely many distinct quasi-isometry classes. The last conclusion, which represents the new feature, shows that negatively curved geometries in dimension 4 exhibit substantially richer structure than those in dimension 3.
ER  -