Families of mutually isospectral Riemannian orbifolds

TL;DR

This paper investigates three arithmetic families of isospectral non-isometric Riemannian orbifolds, providing polynomial bounds on their family sizes relative to the orbifolds' volume.

Contribution

It derives explicit polynomial bounds for the sizes of three distinct families of isospectral orbifolds based on their volume, expanding understanding of spectral geometry.

Findings

Polynomial bounds for family sizes relative to volume

Construction of families via Vigneras' method

Analysis of minimal and maximal arithmetic subgroups

Abstract

In this paper we consider three arithmetic families of isospectral non-isometric Riemannian orbifolds and in each case derive an upper bound for the size of the family which is polynomial as a function of the volume of the orbifolds. The first family that we consider are those constructed by Vigneras' method. The second and third families are those whose covering groups are the minimal covolume arithmetic subgroups and maximal arithmetic subgroups of PGL_2(R)^a x PGL_2(C)^b.