The eigenvalue one property of finite groups, II

TL;DR

This paper proves a conjecture about the existence of eigenvalue one in certain finite group elements acting on odd-dimensional real vector spaces, impacting the classification of flat manifolds.

Contribution

It establishes a sufficient condition for a closed flat manifold to be an $R_{ty}$-manifold by proving a conjecture related to eigenvalues in finite groups.

Findings

Confirmed the conjecture of Dekimpe, De Rock, and Penninckx.

Provided a criterion for identifying $R_{ty}$-manifolds.

Linked eigenvalue properties to geometric manifold classification.

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.