Isospectral Configurations in Euclidean and Hyperbolic Geometry

TL;DR

This paper explores pairs of non-isometric surfaces in Euclidean and hyperbolic geometries that share identical length spectra, providing new constructions especially for hyperbolic surfaces with no common cover.

Contribution

It introduces novel methods for constructing isospectral pairs of hyperbolic surfaces, extending the classical Euclidean examples by Mallows and Clark.

Findings

Existence of isospectral convex dodecagons in Euclidean geometry.

Construction of hyperbolic surface pairs with identical chord length distributions.

Clarification of the relationship between isospectrality and geometric coverings.

Abstract

A number of questions related to the length spectrum of surfaces are discussed and in particular the existence of pairs of surfaces which though not isometric are isospectral. Here by isospectral we mean that a pair of bodies have the same distribution of chord lengths. In the Euclidean setting, we study isospectral convex dodecagons found by Mallows and Clark in the 1970's. Starting from their idea, we give constructions for isospectral pairs of hyperbolic surfaces that have no common cover. Since the work of Mallows and Clark is probably unfamiliar to readers with a background in topology/hyperbolic geometry we include expository material on other related topics about the distribution of chord lengths.