Salem numbers and arithmetic hyperbolic groups

Abstract

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 Salem numbers.

Counting Salem Numbers of Arithmetic Hyperbolic 3-Orbifolds

“…For convenience in this paper we will also allow Salem numbers to have degree 2 (where λ has λ −1 as its only conjugate). We do this so that our definition of Salem numbers aligns with that used in Emery et al (2019).…”

“…Theorem 6 (cf. Emery et al 2019, Theorem 1.6) Let ⊆ Isom(H n ) be an arithmetic lattice, with n odd, of the simplest type defined over a totally real number field L. Let γ be a hyperbolic element of , and let λ = e 2 (γ ) . Then λ is a Salem number which is square-rootable over L.…”

Arithmetic Groups and the Lehmer Conjecture

“…Salem numbers can also be used to obtain a lower bound for the length of closed geodesics in noncompact arithmetic hyperbolic orbifolds of even dimension n. This was done by Emery, Ratcliffe and Tschantz [ERT19]. More precisely, for any integer n ≥ 2, let H n denote hyperbolic n-space, and define…”

Virtually spinning hyperbolic manifolds

“…By definition, any arithmetic subgroup Γ < SO(f, R) contains a welllocated subgroup of finite index. As discussed in § 5.2, when n is even, all arithmetic subgroups commensurable with SO(f, R k ) are contained in SO(f, k), and when n is odd, the group Γ (2) is contained in SO(f, k) (see the proof of [6,Lemma 10]). Now [Γ : Γ (2) ] = |H 1 (Γ, Z/2Z)|, and getting a sharper version of Sullivan's result (i.e., bounding the index) in the general arithmetic setting reduces to consideration of Γ < SO(f, k) and getting an effectively computable constant bounding [Γ : Γ ∩ SO(f, R k )].…”