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

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 $f:M\to M.$ 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

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