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Finiteness of closed arithmetic hyperbolic surface bundles
Abstract
For each g≥ 2, we show that there are finitely many cyclic commensurability classes of closed, fibered arithmetic hyperbolic $3$-manifolds with genus g fiber. More generally, we show that there are only finitely many conjugacy classes of admissible surface subgroups of PSL2(ℂ) whose image is contained in the fundamental group of some arithmetic hyperbolic $3$-manifold. We also give an effectively computable upper bound with effective asymptotic rate on both finiteness statements. This affirms a conjecture of Bowditch, Maclachlan, and Reid.
Related
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BibTeX
@misc{cheethamwest2026finiteness,
title = {{Finiteness of closed arithmetic hyperbolic surface bundles}},
author = {Tam Cheetham-West and Homin Lee and Nicholas Miller},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28188},
eprint = {2609.28188},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28188},
}
RIS
TY - UNPB
TI - Finiteness of closed arithmetic hyperbolic surface bundles
AU - Tam Cheetham-West
AU - Homin Lee
AU - Nicholas Miller
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.28188
UR - https://arxiv.org/abs/2609.28188
AB - For each g≥ 2, we show that there are finitely many cyclic commensurability classes of closed, fibered arithmetic hyperbolic $3$-manifolds with genus g fiber. More generally, we show that there are only finitely many conjugacy classes of admissible surface subgroups of PSL_2(ℂ) whose image is contained in the fundamental group of some arithmetic hyperbolic $3$-manifold. We also give an effectively computable upper bound with effective asymptotic rate on both finiteness statements. This affirms a conjecture of Bowditch, Maclachlan, and Reid.
ER -