Holonomy groups of flat manifolds with $R_\infty$ property

TL;DR

This paper explores the relationship between the holonomy representation of flat manifolds and the $R_ty$ property, establishing conditions under which such manifolds exhibit infinite Reidemeister numbers.

Contribution

It demonstrates that flat manifolds with solvable holonomy groups and a specific irreducible subrepresentation have the $R_ty$ property, advancing understanding of Reidemeister dynamics.

Findings

Holonomy representation influences $R_ty$ property.

Unique odd-degree irreducible subrepresentation implies $R_ty$ property.

Supports conjecture relating holonomy and Reidemeister numbers.

Abstract

Let $M$ be a flat manifold. We say that $M$ has $R_\infty$ property if the Reidemeister number $R(f) = \infty$ for every homeomorphism $f \colon M \to M.$ In this paper, we investigate a relation between the holonomy representation $\rho$ of a flat manifold $M$ and the $R_\infty$ property. In case when the holonomy group of $M$ is solvable we show that, if $\rho$ has a unique $\mathbb{R}$-irreducible subrepresentation of odd degree, then $M$ has $R_\infty$ property. The result is related to conjecture 4.8 from [1]. [1] K. Dekimpe, B. De Rock, P. Penninckx, \emph{The $R_{\infty}$ property for infra-nilmanifolds}, Topol. Methods Nonlinear Anal. 34 (2009), no.2, 353 - 373