On Salem numbers of degree 4 and arithmetic hyperbolic orbifolds

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 $\mathbb{Q}^{\times}/\mathbb{Q}^{\times 2}$.