Salem numbers and commensurability classes of arithmetic hyperbolic manifolds

TL;DR

This work links Salem numbers to lengths of closed geodesics in arithmetic hyperbolic manifolds by showing that, for any Salem number $λ$ and a suitable totally real field $k$, there exist infinitely many pairwise incommensurable arithmetic lattices in $\mathrm{Isom}(\mathbb{H}^n)$ over $k$ that realize $λ$ via a geodesic of length $\log λ$ in every allowable dimension $n$. The authors build on Bayer–Fluckiger’s Hasse principle for isometries with prescribed eigenvalues and Maclachlan’s commensurability parametrization, then leverage class field theory to construct infinitely many local-data patterns that yield distinct commensurability classes. A precise finite set of necessary and sufficient conditions is given for realizability of $λ$ in a given commensurability class, depending on the degree gap $n+1-\deg_k(λ)$ and parity constraints. The resulting infinite families broaden the known realizations of Salem numbers in arithmetic hyperbolic geometry and deepen the connection between algebraic number theory and geometric length spectra. The methods provide a framework to generate many incommensurable arithmetic manifolds with prescribed spectral data, highlighting the power of local-global invariants in distinguishing commensurability classes.

Abstract

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

Theorems & Definitions (46)

• Theorem 1.1
• Theorem 1.2
• Theorem A
• Theorem 2.1: Classification of quaternion algebras
• Proposition 2.2
• Lemma 2.3
• Theorem 3.1
• Definition 3.2
• Definition 3.3
• Proposition 3.4