Salem numbers and arithmetic hyperbolic groups

@article{Emery2019SalemNA,
  title={Salem numbers and arithmetic hyperbolic groups},
  author={Vincent Emery and John G. Ratcliffe and Steven T. Tschantz},
  journal={Transactions of the American Mathematical Society},
  year={2019}
}
In this paper we prove that there is a direct relationship between Salem numbers and translation lengths of hyperbolic elements of arithmetic hyperbolic groups that are determined by a quadratic form over a totally real number field. As an application we determine a sharp lower bound for the length of a closed geodesic in a noncompact arithmetic hyperbolic n-orbifold for each dimension n. We also discuss a "short geodesic conjecture", and prove its equivalence with "Lehmer's conjecture" for… 
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Methods Citations
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14 Citations

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References

SHOWING 1-10 OF 31 REFERENCES
The arithmetic and geometry of Salem numbers
A Salem number is a real algebraic integer, greater than 1, with the property that all of its conjugates lie on or within the unit circle, and at least one conjugate lies on the unit circle. In this…
Commensurability classes of discrete arithmetic hyperbolic groups
In this paper we show that the commensurability classes of discrete arithmetic subgroups of Isom.H/ of the simplest type, i.e., those coming from quadratic forms of signature .n; 1/ over totally real…
Number Theory and Polynomials: The Mahler measure of algebraic numbers: a survey
A survey of results for Mahler measure of algebraic numbers, and one-variable polynomials with integer coefficients is presented. Related results on the maximum modulus of the conjugates (`house') of…
Arithmetic of Hyperbolic Manifolds
By a hyperbolic 3-manifold we mean a complete orientable hyperbolic 3-manifold of finite volume, that is a quotient H/Γ with Γ ⊂ PSL2C a discrete subgroup of finite covolume (here briefly “a Kleinian…
The Geometry of Discrete Groups
Describing the geometric theory of discrete groups and the associated tesselations of the underlying space, this work also develops the theory of Mobius transformations in n-dimensional Euclidean…
Rational Quadratic Forms
Eigenvalue fields of hyperbolic orbifolds
In this paper, we prove that if F is a non-elementary subgroup of O o (n,1,R), with n > 2, then the eigenvalue field of Γ has infinite degree over Q.
Parametrizing Fuchsian Subgroups of the Bianchi Groups
Let dbe a positive square-free integer and let Od denote the ring of integers in . The groups PSL2(Od) are collectively known as the Bianchi groups and have been widely studied from the viewpoints of…
The Book of Involutions
This monograph yields a comprehensive exposition of the theory of central simple algebras with involution, in relation with linear algebraic groups. It aims to provide the algebra-theoretic…
The Bianchi groups are separable on geometrically finite subgroups
Let d be a square free positive integer and Od the ring of integers in Q( p id). The main result of this paper is that the groups PSL(2;Od) are
...
...