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Arithmetic hyperbolic manifolds with bounded rank

  • Matthew Zevenbergen
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Abstract

We prove that closed arithmetic hyperbolic n-manifolds of bounded rank have bounded adjoint trace field degree, and consequently have lower bounded injectivity radius. From this, we obtain a number of corollaries. In particular, we obtain that closed arithmetic hyperbolic $3$-manifolds of bounded rank are contained in finitely many commensurability classes, and describe these commensurability classes. Additionally, it follows that for closed arithmetic hyperbolic $3$-manifolds, Heegaard genus is equal to rank for all but finitely many manifolds of each rank.

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Cite

BibTeXRIS
BibTeX
@misc{zevenbergen2026arithmetic,
  title = {{Arithmetic hyperbolic manifolds with bounded rank}},
  author = {Matthew Zevenbergen},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.18841},
  eprint = {2610.18841},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.18841},
}
RIS
TY  - UNPB
TI  - Arithmetic hyperbolic manifolds with bounded rank
AU  - Matthew Zevenbergen
PY  - 2026
DA  - 2026/10/13
PB  - arXiv
DO  - 10.48550/arxiv.2610.18841
UR  - https://arxiv.org/abs/2610.18841
AB  - We prove that closed arithmetic hyperbolic n-manifolds of bounded rank have bounded adjoint trace field degree, and consequently have lower bounded injectivity radius. From this, we obtain a number of corollaries. In particular, we obtain that closed arithmetic hyperbolic $3$-manifolds of bounded rank are contained in finitely many commensurability classes, and describe these commensurability classes. Additionally, it follows that for closed arithmetic hyperbolic $3$-manifolds, Heegaard genus is equal to rank for all but finitely many manifolds of each rank.
ER  -