Salem numbers and commensurability classes of arithmetic hyperbolic manifolds

TL;DR

This paper demonstrates the existence of infinitely many commensurability classes of arithmetic hyperbolic manifolds containing geodesics of specific lengths related to Salem numbers, linking number theory and geometric topology.

Contribution

It establishes a new connection between Salem numbers and the geometry of arithmetic hyperbolic manifolds, showing how Salem numbers determine geodesic lengths within these manifolds.

Findings

Existence of infinitely many commensurability classes with prescribed geodesic lengths

Link between Salem numbers and hyperbolic manifold geometry

Construction of manifolds over specific totally real fields

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 $\log\lambda$.