Hyperbolic manifolds and pseudo-arithmeticity

Emery, Vincent C.; Mila, Olivier

doi:10.48550/arxiv.1810.12837

Cited by 3 publications

(7 citation statements)

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“…Remark 5.4. This gives another proof of Theorem 2.7 in [4]. If the pieces are fundamental pieces with canonical gluing isometries, Theorem 4.1 even permits an exact computation of the trace field.…”

Section: Corollaries and Auxiliary Resultsmentioning

confidence: 87%

“…PO f i ) is the trace field (resp. the ambient group) of the arithmetic manifold M i is a piece of, see [4,Lem. 2.5].…”

Section: Corollaries and Auxiliary Resultsmentioning

confidence: 99%

“…, M r . As shown for example in [4], the orbifold ∆ 5 zH 5 (and thus M ) has trace field Qp ? 2q and ambient group PO f with f " ´x2 0 `x2 1 `¨¨¨`x 2 5 .…”

Section: Examples and Applicationsmentioning

confidence: 91%

“…Since PO f is adjoint, this implies σpφq " φ or σpφq " φρ (where ρ is the reflection at R 1 ). Moreover, since ρ P PO f pkq by Lemma 2.1 of [4], we have σpφρq " σpφqρ and thus σ acts on the set tφ, φρu.…”

Section: Proof Let ămentioning

confidence: 96%

“…2q and ambient group PO f with f " ´x2 0 `x2 1 `¨¨¨`x 2 5 . By Lemma 3.2, the trace field of the pieces (and thus of the M i by Lemma 2.5 of [4]) is Q or Qp ? 2q.…”

Section: Examples and Applicationsmentioning

confidence: 99%

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The trace field of hyperbolic gluings

Mila¹

2019

Preprint

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We determine the adjoint trace field of gluings of general hyperbolic manifolds. This provides a new method to prove the nonarithmeticity of gluings, which can be applied to the classical construction of Gromov and Piatetski-Shapiro (and generalizations) as well as certain gluings of pieces of commensurable arithmetic manifolds. As an application we give many new examples of nonarithmetic gluings and prove that the unique nonarithmetic Coxeter 5-simplex is not commensurable to any gluing of arithmetic pieces.

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Section: Corollaries and Auxiliary Resultsmentioning

confidence: 87%

“…PO f i ) is the trace field (resp. the ambient group) of the arithmetic manifold M i is a piece of, see [4,Lem. 2.5].…”

Section: Corollaries and Auxiliary Resultsmentioning

confidence: 99%

“…, M r . As shown for example in [4], the orbifold ∆ 5 zH 5 (and thus M ) has trace field Qp ? 2q and ambient group PO f with f " ´x2 0 `x2 1 `¨¨¨`x 2 5 .…”

Section: Examples and Applicationsmentioning

confidence: 91%

Section: Proof Let ămentioning

confidence: 96%

“…2q and ambient group PO f with f " ´x2 0 `x2 1 `¨¨¨`x 2 5 . By Lemma 3.2, the trace field of the pieces (and thus of the M i by Lemma 2.5 of [4]) is Q or Qp ? 2q.…”

Section: Examples and Applicationsmentioning

confidence: 99%

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The trace field of hyperbolic gluings

Mila¹

2019

Preprint

Self Cite

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Finiteness of maximal geodesic submanifolds in hyperbolic hybrids

et al. 2021

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We show that large classes of non-arithmetic hyperbolic n-manifolds, including the hybrids introduced by Gromov and Piatetski-Shapiro and many of their generalizations, have only finitely many finite-volume immersed totally geodesic hypersurfaces. In higher codimension, we prove finiteness for geodesic submanifolds of dimension at least 2 that are maximal, i.e., not properly contained in a proper geodesic submanifold of the ambient n-manifold. The proof is a mix of structure theory for arithmetic groups, dynamics, and geometry in negative curvature.

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