Salem numbers and commensurability classes of arithmetic hyperbolic manifolds

Abstract

In this article we show that given a Salem number $\lambda$, a totally real number field $k\subseteq\mathbb{Q}(\lambda+\lambda^{-1})$, and a positive integer $n\geq\mathrm{deg}_k(\lambda)-1$, there exist infinitely many commensurability classes of arithmetic hyperbolic $n$-manifolds defined over $k$ which contain a geodesic of length $\log\lambda$.