On Salem numbers of degree 4 and arithmetic hyperbolic orbifolds
Cayo Dória, Plinio G. P. Murillo
TL;DR
The paper addresses whether a single classical arithmetic lattice of the first type can realize all Salem numbers up to a given degree as translation lengths of closed geodesics. It constructs a nonuniform arithmetic hyperbolic 6-orbifold $\mathcal{O}$ that realizes all Salem numbers of degree $2$ and all square-rootable Salem numbers of degree $4$, via a Clifford-algebraic embedding into $\mathrm{SO}_{q_6}$ and the Vahlen group framework. It proves that dimension $6$ is minimal for this universal realization and establishes a discriminant–determinant relationship in an appendix, including a geometric method to produce degree-$2m$ Salem numbers with prescribed discriminants. These results connect Salem-number arithmetic to hyperbolic geometry through Clifford algebras, Witt invariants, and explicit lattice constructions, expanding the understanding of length spectra in arithmetic hyperbolic orbifolds. The findings have implications for the distribution of geodesic lengths in high-dimensional arithmetic manifolds and highlight the role of square-rootable Salem numbers in odd- and even-dimensional realizations.
Abstract
In this article, we construct an arithmetic hyperbolic $6-$orbifold $\mathcal{O}$ such that, any square-rootable Salem number of degree at most $4$ over $\mathbb{Q}$ is realized as the exponential of the length of a closed geodesic in $\mathcal{O}$. We also prove that $n=6$ is the minimal dimension among arithmetic hyperbolic orbifolds of the first type where it can be obtained. In an appendix, we establish a general relation between the discriminant of a Salem number and the determinant of a quadratic space which realizes it. In particular, for any $m,d>0$ we present a geometric proof of the existence of Salem numbers of degree $2m$ with discriminant $(-1)^{m+1}d$ in $\mathbb{Q}^{\times}/\mathbb{Q}^{\times 2}$.