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

TL;DR

This paper investigates the algebraic and transcendental properties of the length spectra of locally symmetric spaces, establishing a dichotomy under Schanuel's conjecture related to their length-commensurability and field extensions.

Contribution

It proves, assuming Schanuel's conjecture, that arithmetically defined locally symmetric spaces are either length-commensurable or have highly transcendental length spectrum fields.

Findings

If length spectra are commensurable, their associated fields are proportional.

Otherwise, the combined field has infinite transcendence degree over each individual field.

The results depend on Schanuel's conjecture from transcendental number theory.

Abstract

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).