Holonomy groups of flat manifolds with the \(R_\infty\) property. (Q2866775)

Given a manifold \(M\), the \(R_\infty\) property is defined as follows. A homeomorphism \(f\colon M\to M\) induces an automorphism \(f_\#\colon\Gamma\to\Gamma\) of the fundamental group \(\Gamma=\pi_1(M)\). Two elements \(\alpha,\beta\in\Gamma\) are called \(f_\#\)-conjugate if there exists \(\gamma\in\Gamma\) such that \(\beta=\gamma\alpha f_\#(\gamma)^{-1}\). The \(f_\#\)-conjugacy class of \(\alpha\) containing all such elements \(\beta\) is called a Reidemeister class of \(f\). The number of Reidemeister classes is called the Reidemeister number \(R(f)\) of \(f\). The manifold \(M\) has the \(R_\infty\) property if \(R(f)\) is infinite for every homeomorphism \(f\) of \(M\).NEWLINENEWLINE The authors study the relation between the holonomy representation \(\rho\colon G\to\text{GL}(n,\mathbb Z)\) of a flat manifold \(M\) and the \(R_\infty\) property.NEWLINENEWLINE The main result of the paper is that if the holonomy group \(G\) of \(M\) is solvable and there exists a \(\mathbb Q\)-irreducible \(\mathbb Q\)-subrepresentation \(\rho'\colon G\to\text{GL}(n',\mathbb Z)\) of \(\rho\) of odd dimension such that \(\rho'(G)\) is not \(\mathbb Q\)-conjugate to \(\widetilde\rho(G)\) for any other \(\mathbb Q\)-subrepresentation \(\widetilde\rho\) of \(\rho\), then \(M\) has the \(R_\infty\) property. This results is related to Conjecture 4.8 of \textit{K. Dekimpe} et al., [Topol. Methods Nonlinear Anal. 34, No. 2, 353-373 (2009; Zbl 1200.55003)].