Salem numbers and commensurability classes of arithmetic hyperbolic manifolds

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 $n\ge {\mathrm{d}\mathrm{e}\mathrm{g}}_k(\lambda )-1$, there exist infinitely many commensurability classes of arithmetic hyperbolic $n$-manifolds defined over $k$ which contain a geodesic of length $\log \lambda$.