The Sunada construction and the simple length spectrum

TL;DR

This paper investigates hyperbolic surfaces created through the Sunada construction, revealing that they can share the same length spectrum but differ in their simple length spectrum, highlighting limitations in spectral geometry.

Contribution

It demonstrates that Sunada-constructed hyperbolic surfaces can be iso-length spectral without being simple iso-length spectral, clarifying spectral distinctions.

Findings

Sunada construction can produce hyperbolic surfaces with identical length spectra

Such surfaces may differ in their simple length spectra

The result clarifies limitations of spectral geometry methods

Abstract

We show that certain families of iso-length spectral hyperbolic surfaces obtained via the Sunada construction are not generally simple iso-length spectral.