Irrationality of the Length Spectrum

TL;DR

The paper proves that for a non-elementary Fuchsian group $\Gamma$, the length spectrum $L(\Gamma)$ contains two lengths that are linearly independent over $\mathbb{Q}$, strengthening the classical non-arithmeticity result of Dal'Bo. The authors provide an elementary proof by connecting the length data to radicals in finitely generated extensions of $\mathbb{Q}$ and the trace field $k\Gamma$, showing that if all lengths were rational multiples of a fixed length, one would obtain a contradiction with non-arithmeticity. A central step uses the relation $|\mathrm{tr}(\gamma)|=e^{\ell(\gamma)/2}+e^{-\ell(\gamma)/2}$ and a finiteness result on radicals (via a key lemma about degrees) to force a bound on denominators of lengths and hence discreteness of the length subgroup. The argument, which leverages elementary number theory alongside trace-field considerations, extends to hyperbolic isometries in $\mathrm{SO}(n,1)^+$. The work connects to existing results of Prasad–Rapinchuk and situates the approach within arithmetic aspects of length spectra and heights.

Abstract

It is a classical result of Dal'Bo that the length spectrum of a non-elementary Fuchsian group is non-arithmetic, namely, it generates a dense additive subgroup of $\mathbb{R}$. In this note we provide an elementary proof of an extension of this theorem: a non-elementary Fuchsian group contains two elements whose lengths are linearly independent over $\mathbb{Q}$, reproving a result of Prasad and Rapinchuk.

Theorems & Definitions (15)

• Definition 1.1
• Theorem 1.2: Non-arithmeticity of the length spectrum, Dal'Bo1
• Theorem 1.3
• Definition 2.1
• Lemma 2.2
• proof
• Proposition 2.3
• proof : Proof of \ref{['Theorem: Main theorem']}
• Remark 2.4
• Proposition 3.1