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The fundamental group of a spherical space form is not audible

Mauro Colantonio, Emilio A. Lauret

TL;DR

The paper resolves whether the fundamental group of spherical space forms is audible by proving the existence of infinitely many pairs of isospectral spherical space forms with non-isomorphic fundamental groups, constructed via fixed-point-free Type I groups $ extΓ_d(m,n,r)$ and careful representation-theoretic assembly. It establishes audible constraints on Type I invariants, showing that parameters $(m,n,d)$ and certain gcds are determined by the spectrum, and it demonstrates that Type I can be audible under specific conditions. The authors also present a constructive method to produce strong isospectrality between quotients of spheres by non-isomorphic Type I groups, and support the theory with computational searches and a detailed analysis of fixed-point-free representations. The work highlights both the limits of spectral data in recovering topology and structure for spherical space forms, and the circumstances under which the fundamental group can be heard, offering new avenues for isospectrality theory and computational verification.

Abstract

We revisit the problem of isospectral spherical space forms with non-cyclic fundamental groups after the works by Ikeda, Gilkey and Wolf. We find the first pair of spherical space forms with non-isomorphic fundamental groups and the same Laplace spectrum. This shows that the isomorphism class of the fundamental group is not audible among spherical space forms. We also found several instances where one can hear the fundamental group of a spherical space form (among spherical space forms).

The fundamental group of a spherical space form is not audible

TL;DR

The paper resolves whether the fundamental group of spherical space forms is audible by proving the existence of infinitely many pairs of isospectral spherical space forms with non-isomorphic fundamental groups, constructed via fixed-point-free Type I groups and careful representation-theoretic assembly. It establishes audible constraints on Type I invariants, showing that parameters and certain gcds are determined by the spectrum, and it demonstrates that Type I can be audible under specific conditions. The authors also present a constructive method to produce strong isospectrality between quotients of spheres by non-isomorphic Type I groups, and support the theory with computational searches and a detailed analysis of fixed-point-free representations. The work highlights both the limits of spectral data in recovering topology and structure for spherical space forms, and the circumstances under which the fundamental group can be heard, offering new avenues for isospectrality theory and computational verification.

Abstract

We revisit the problem of isospectral spherical space forms with non-cyclic fundamental groups after the works by Ikeda, Gilkey and Wolf. We find the first pair of spherical space forms with non-isomorphic fundamental groups and the same Laplace spectrum. This shows that the isomorphism class of the fundamental group is not audible among spherical space forms. We also found several instances where one can hear the fundamental group of a spherical space form (among spherical space forms).
Paper Structure (13 sections, 17 theorems, 51 equations, 1 table, 1 algorithm)

This paper contains 13 sections, 17 theorems, 51 equations, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Spherical space forms
  4. Type I groups
  5. Audible properties
  6. The set of orders
  7. Hearing parameters
  8. Hearing the type
  9. Counterexamples
  10. Fixed point free representations
  11. Isospectral construction
  12. Computational results
  13. Open questions

Key Result

Theorem 1.2

Let $\Gamma$ be a non-cyclic fixed point free Type I group. Given two irreducible real representations $\rho_1,\rho_2$ of $\Gamma$ such that $\rho_1(\Gamma)$ and $\rho_2(\Gamma)$ act freely on the unit sphere $S^d$ (they have necessarily the same dimension $d+1$), then $\mathop{\mathrm{Spec}}\nolimi

Theorems & Definitions (38)

  • Remark 1.1
  • Theorem 1.2: Ikeda83
  • Remark 1.3
  • Theorem 1.5
  • Remark 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Lemma 2.1
  • Lemma 2.2
  • Remark 2.3
  • ...and 28 more