Representation equivalent Bieberbach groups and strongly isospectral flat manifolds

TL;DR

This paper proves that strongly isospectral flat manifolds derived from Bieberbach groups imply the groups are representation equivalent, linking spectral properties to algebraic group structures.

Contribution

It establishes a connection between spectral isospectrality of flat manifolds and the representation equivalence of their Bieberbach groups.

Findings

Strongly isospectral flat manifolds imply representation equivalence of Bieberbach groups.

The result links geometric spectral data with algebraic group representations.

Provides a new criterion for comparing Bieberbach groups via spectral analysis.

Abstract

Let $\Gamma_1$ and $\Gamma_2$ be Bieberbach groups contained in the full isometry group $G$ of $\mathbb{R}^n$. We prove that if the compact flat manifolds $\Gamma_1\backslash\mathbb{R}^n$ and $\Gamma_2\backslash\mathbb{R}^n$ are strongly isospectral then the Bieberbach groups $\Gamma_1$ and $\Gamma_2$ are representation equivalent, that is, the right regular representations $L^2(\Gamma_1\backslash G)$ and $L^2(\Gamma_2\backslash G)$ are unitarily equivalent.