Non-Pseudo-Arithmetic Hyperbolic 4-Manifolds
Abstract
We give three explicit finite-volume hyperbolic Coxeter lattices in dimension four whose adjoint trace fields are not totally real. They are the reflection groups of the Ma--Zheng polytopes P11,6, P11,7, P11,8. The obstruction is contained in two cyclic products of the doubled unit-normal Gram matrix A=2G. Those products recover elements s=√5 and u satisfying (s−1)u2+2(s−3)u+s−7=0. After the embedding s↦−√5, this quadratic has discriminant 8(1−√5)<0. The cyclic-product field therefore has a complex place. Vinberg's identification of that field with the adjoint trace field and the Emery--Mila total-reality obstruction imply that the three lattices are not pseudo-arithmetic. Torsion-free orientation-preserving finite-index subgroups yield orientable noncompact finite-volume hyperbolic $4$-manifolds with the same obstruction. Exact computations check the census data and provide independent field and finite-volume certificates. A pinned Lean~4 development verifies the radical identity and the number-field obstruction. It also checks the lattice and manifold implications conditional on explicit geometric and literature hypotheses.
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BibTeX
@misc{kolpakov2026nonpseudoarithmetic,
title = {{Non-Pseudo-Arithmetic Hyperbolic 4-Manifolds}},
author = {Alexander Kolpakov},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05795},
eprint = {2611.05795},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05795},
}
RIS
TY - UNPB
TI - Non-Pseudo-Arithmetic Hyperbolic 4-Manifolds
AU - Alexander Kolpakov
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.05795
UR - https://arxiv.org/abs/2611.05795
AB - We give three explicit finite-volume hyperbolic Coxeter lattices in dimension four whose adjoint trace fields are not totally real. They are the reflection groups of the Ma--Zheng polytopes P_11,6, P_11,7, P_11,8. The obstruction is contained in two cyclic products of the doubled unit-normal Gram matrix A=2G. Those products recover elements s=√5 and u satisfying (s−1)u^2+2(s−3)u+s−7=0. After the embedding s↦−√5, this quadratic has discriminant 8(1−√5)<0. The cyclic-product field therefore has a complex place. Vinberg's identification of that field with the adjoint trace field and the Emery--Mila total-reality obstruction imply that the three lattices are not pseudo-arithmetic. Torsion-free orientation-preserving finite-index subgroups yield orientable noncompact finite-volume hyperbolic $4$-manifolds with the same obstruction. Exact computations check the census data and provide independent field and finite-volume certificates. A pinned Lean~4 development verifies the radical identity and the number-field obstruction. It also checks the lattice and manifold implications conditional on explicit geometric and literature hypotheses.
ER -