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The eigenvalue one property of finite groups, I

Gerhard Hiss, Rafał Lutowski

TL;DR

This work proves the DDRP eigenvalue-one conjecture by establishing the eigenvalue-one property for all finite groups acting irreducibly on odd-dimensional real vector spaces, combining a restriction method and a large-degree criterion. The authors reduce to finite simple groups and treat sporadic, alternating, and Lie-type groups, employing Harish-Chandra theory, BN-pairs, and automorphism analysis to rule out minimal counterexamples. As a consequence, any closed flat manifold with a holonomy representation containing an odd-degree irreducible real subrepresentation satisfying a multiplicity-one condition is an $R_ fty$-manifold. The results bridge deep representation-theoretic tools with geometric implications for Reidemeister numbers and fixed-point theory, and provide a comprehensive framework for tackling eigenvalue problems across finite group families.

Abstract

We prove a conjecture of Dekimpe, De Rock and Penninckx concerning the existence of eigenvalues one in certain elements of finite groups acting irreducibly on a real vector space of odd dimension. This yields a sufficient condition for a closed flat manifold to be an $R_{\infty}$-manifold.

The eigenvalue one property of finite groups, I

TL;DR

This work proves the DDRP eigenvalue-one conjecture by establishing the eigenvalue-one property for all finite groups acting irreducibly on odd-dimensional real vector spaces, combining a restriction method and a large-degree criterion. The authors reduce to finite simple groups and treat sporadic, alternating, and Lie-type groups, employing Harish-Chandra theory, BN-pairs, and automorphism analysis to rule out minimal counterexamples. As a consequence, any closed flat manifold with a holonomy representation containing an odd-degree irreducible real subrepresentation satisfying a multiplicity-one condition is an -manifold. The results bridge deep representation-theoretic tools with geometric implications for Reidemeister numbers and fixed-point theory, and provide a comprehensive framework for tackling eigenvalue problems across finite group families.

Abstract

We prove a conjecture of Dekimpe, De Rock and Penninckx concerning the existence of eigenvalues one in certain elements of finite groups acting irreducibly on a real vector space of odd dimension. This yields a sufficient condition for a closed flat manifold to be an -manifold.

Paper Structure

This paper contains 30 sections, 39 theorems, 43 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Motivation, the conjecture, and the main result
  3. Two methods
  4. Survey of the paper
  5. Notation and preliminaries
  6. Numbers
  7. Graphs
  8. Groups
  9. Matrices
  10. Characters and modules
  11. The reduction to finite simple groups
  12. The restriction method
  13. The minimal counterexamples
  14. The $E1$-property for the simple sporadic and alternating groups
  15. General notation
  16. ...and 15 more sections

Key Result

Theorem 1.1.1

DDRP Let $M$ be a closed flat manifold with holonomy representation (Rho). Suppose there is a $\mathbb{Z}$-sub-re-pre-sent-ati-on $\rho \colon G \rightarrow \text{\rm GL}_d( \mathbb{Z} )$, such that $\rho^{\mathbb{Q}}$ is irreducible and of multiplicity one as a composition factor of $\gamma^{\math

Figures (1)

  • Figure 5.1: The Dynkin diagrams of $A_{d-1}$, $E_6$ and $D_4$

Theorems & Definitions (79)

  • Theorem 1.1.1
  • Conjecture 1.1.2
  • Definition 1.1.3
  • Theorem 1.1.5
  • Corollary 1.1.6
  • Lemma 2.5.1
  • proof
  • Lemma 2.5.2
  • proof
  • Lemma 2.5.3
  • ...and 69 more