Iso-length-spectral hyperbolic surface amalgams

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 non-commensurable pair with the same weak length spectrum, the length set without multiplicity.