Arithmetic groups and the Lehmer conjecture

Pham, Lam; Thilmany, François

doi:10.1007/s11856-021-2229-7

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Conjecture (Salem Conjecture)1

Theorem (Pham-thilmany[pt21]1

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2022

2023

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Cited by 2 publications

(2 citation statements)

References 26 publications

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“…The main new content of Theorem A is that the Salem conjecture implies the Margulis conjecture for any simple Lie group . The fact that Conjecture 1.3 follows from Conjecture 1.1 for any isotropic almost simple R-group already follows from the main theorem of the second-named author and F. Thilmany [PT21].…”

Section: Conjecture (Salem Conjecture)mentioning

confidence: 86%

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Bottom of the Length Spectrum of Arithmetic Orbifolds

Frączyk¹,

Pham²

2022

Preprint

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A. We prove that cocompact arithmetic lattices in a simple Lie group are uniformly discrete if and only if the Salem numbers are uniformly bounded away from 1. We also prove an analogous result for semisimple Lie groups. Finally, we shed some light on the structure of the bottom of the length spectrum of an arithmetic orbifold Γ\ by showing the existence of a positive constant ( ) > 0 such that squares of lengths of closed geodesics shorter than must be pairwise linearly dependent over Q.

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Section: Conjecture (Salem Conjecture)mentioning

confidence: 86%

“…Note that in the other direction, the main theorem of [PT21] shows that the Margulis conjecture for for all signatures ( , ) where 0 ≤ ≤ 2 and 1 ≤ + 2 ≤ 1 + 2 2 implies the Lehmer conjecture (1.7) for the same set of signatures.…”

Section: Theorem (Pham-thilmany[pt21]mentioning

confidence: 99%

Bottom of the Length Spectrum of Arithmetic Orbifolds

Frączyk¹,

Pham²

2022

Preprint

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Bottom of the length spectrum of arithmetic orbifolds

Fráczyk

Pham

2023

Trans. Amer. Math. Soc.

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We prove that cocompact arithmetic lattices in a simple Lie group are uniformly discrete if and only if Salem numbers are uniformly bounded away from 1 1 . We also prove an analogous result for semisimple Lie groups. Finally, we shed some light on the structure of the bottom of the length spectrum of an arithmetic orbifold Γ ∖ X \Gamma \backslash X by showing the existence of a positive constant δ ( X ) > 0 \delta (X)>0 such that squares of lengths of closed geodesics shorter than δ \delta must be pairwise linearly dependent over Q \mathbb {Q} .

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Arithmetic groups and the Lehmer conjecture

Cited by 2 publications

References 26 publications

Bottom of the Length Spectrum of Arithmetic Orbifolds

Bottom of the Length Spectrum of Arithmetic Orbifolds

Bottom of the length spectrum of arithmetic orbifolds

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