On the fields generated by the lengths of closed geodesics in locally symmetric spaces

This paper is the next installment of our analysis of length-commensurable locally symmetric spaces begun in Publ. math. IHES 109(2009), 113-184. For a Riemannian manifold $M$, we let $L(M)$ be the weak length spectrum of $M$, i.e. the set of lengths of all closed geodesics in $M$, and let $\mathcal{F}(M)$ denote the subfield of $\mathbb{R}$ generated by $L(M)$. Let now $M_i$ be an arithmetically defined locally symmetric space associated with a simple algebraic $\mathbb{R}$-group $G_i$ for $i = 1, 2$. Assuming Schanuel's conjecture from transcendental number theory, we prove (under some minor technical restrictions) the following dichotomy: either $M_1$ and $M_2$ are length-commensurable, i.e. $\mathbb{Q} \cdot L(M_1) = \mathbb{Q} \cdot L(M_2)$, or the compositum $\mathcal{F}(M_1)\mathcal{F}(M_2)$ has infinite transcendence degree over $\mathcal{F}(M_i)$ for at least one $i = 1$ or $2$ (which means that the sets $L(M_1)$ and $L(M_2)$ are very different).