Small isospectral and nonisometric orbifolds of dimension 2 and 3

TL;DR

This paper constructs small pairs of 2- and 3-dimensional orbifolds and manifolds that are Laplace isospectral yet nonisometric, using arithmetic groups and revisiting Vigneras's construction.

Contribution

It provides explicit examples of small isospectral nonisometric orbifolds and manifolds in low dimensions through an arithmetic approach.

Findings

Existence of small isospectral nonisometric orbifolds in dimension 2 and 3

Construction based on arithmetic Fuchsian and Kleinian groups

Revisits and extends Vigneras's original construction

Abstract

Revisiting a construction due to Vigneras, we exhibit small pairs of orbifolds and manifolds of dimension 2 and 3 arising from arithmetic Fuchsian and Kleinian groups that are Laplace isospectral (in fact, representation equivalent) but nonisometric.