The $R_{\infty}$-property of flat manifolds: Toward the eigenvalue one property of finite groups

TL;DR

This paper explores the $R_{}$-property of flat manifolds, proposing a conjecture related to the eigenvalue one property of finite groups and outlining key steps toward its proof.

Contribution

It introduces a new conjecture connecting flat manifolds and finite group eigenvalue properties, providing an extended account of related theoretical developments.

Findings

Proposed a conjecture on the $R_{}$-property of flat manifolds

Outlined major steps toward proving the conjecture

Connected eigenvalue one property with finite group theory

Abstract

We introduce a conjecture of Dekimpe, De Rock and Penninckx and sketch some major steps in its proof. This text presents an extended account of the talk of the first author given at the conference Ischia Group Theory 2024.