Iso-length-spectral Hyperbolic Surface Amalgams

TL;DR

This paper constructs new examples of iso-length-spectral hyperbolic surfaces and amalgams, including non-homeomorphic and noncommensurable pairs, extending Sunada's and Buser's earlier work.

Contribution

It generalizes Buser's combinatorial construction to produce iso-length-spectral surface amalgams that are not isometric, including non-homeomorphic and noncommensurable pairs.

Findings

Constructed iso-length-spectral surface amalgams that are not isometric

Found both homeomorphic and non-homeomorphic pairs

Created a noncommensurable pair with the same weak length spectrum

Abstract

Two negatively curved metric spaces are iso-length-spectral if they have the same multisets of lengths of closed geodesics. A well-known paper by Sunada provides a systematic way of constructing iso-length-spectral surfaces that are not isometric. In this paper, we construct examples of iso-length-spectral surface amalgams that are not isometric, generalizing Buser's combinatorial construction of Sunada's surfaces. We find both homeomorphic and non-homeomorphic pairs. Finally, we construct a noncommensurable pair with the same weak length spectrum, the length set without multiplicity.