On volumes of quasi-arithmetic hyperbolic lattices

Emery, Vincent

doi:10.1007/s00029-017-0308-8

Cited by 7 publications

(14 citation statements)

References 25 publications

(36 reference statements)

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“…For n even the result readily follows from the Gauss-Bonnet theorem, and Theorem 1.12 has no interest there (but Theorem 3.5 presumably has). For manifolds obtained by gluing the result was already known; see [7,Sect. 1.4].…”

Section: The Case Of Glued Manifoldsmentioning

confidence: 90%

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Hyperbolic manifolds and pseudo-arithmeticity

Emery

Mila²

2021

Trans. Amer. Math. Soc. Ser. B

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We introduce and motivate a notion of pseudo-arithmeticity, which possibly applies to all lattices in PO ⁡ ( n , 1 ) \operatorname {PO}(n,1) with n > 3 n>3 . We further show that under an additional assumption (satisfied in all known cases), the covolumes of these lattices correspond to rational linear combinations of special values of L L -functions.

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Section: The Case Of Glued Manifoldsmentioning

confidence: 90%

“…1.4]. Note also that for quasi-arithmetic lattices (which corresponds to the case r = 0 above) the result is proved in [7,Theorem 1.3], and the condition "of the first type" is superfluous.…”

Section: The Case Of Glued Manifoldsmentioning

confidence: 98%

Hyperbolic manifolds and pseudo-arithmeticity

Emery

Mila²

2021

Trans. Amer. Math. Soc. Ser. B

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“…In this section we will prove Proposition 5.5 and use it (as well as results from [Em17]) to deduce Theorem 5.2.…”

Section: Proofsmentioning

confidence: 97%

“…The goal of this section is to prove Theorem 5.2. We will start by computing the homology of the k-points PO • f (k) of the algebraic groups PO • f , and deduce the theorem using results of [Em17].…”

Section: Volumes Of Pseudo-arithmetic Manifoldsmentioning

confidence: 99%

The Trace Field of Hyperbolic Gluings

Mila

2020

International Mathematics Research Notices

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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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“…If Γ above is a lattice that satisfies conditions (a) and (b), but not necessarily (c), then we call Γ a quasi-arithmetic lattice. If only (a) and (b) are satisfied while (c) fails then Γ is called a properly quasi-arithmetic lattice [4].…”

Section: Introductionmentioning

confidence: 99%

Infinitely many quasi-arithmetic maximal reflection groups

Dotti¹,

Kolpakov²

2021

Preprint

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In contrast to the fact that there are only finitely many maximal arithmetic reflection groups acting on the hyperbolic space H n , n ≥ 2, we show that (a) one can produce infinitely many maximal quasi-arithmetic reflection groups acting on H 2 ; (b) they admit infinitely many different fields of definition; (c) the degrees of their fields of definition are unbounded. However, for n ≥ 14 the technique initially developed by Vinberg shows that there are still finitely many fields of definitions in the quasi-arithmetic case.

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On volumes of quasi-arithmetic hyperbolic lattices

Cited by 7 publications

References 25 publications

Hyperbolic manifolds and pseudo-arithmeticity

Hyperbolic manifolds and pseudo-arithmeticity

The Trace Field of Hyperbolic Gluings

Infinitely many quasi-arithmetic maximal reflection groups

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