Holonomy groups of flat manifolds with the $R_{\infty} $ property
Volume 223 / 2013
Fundamenta Mathematicae 223 (2013), 195-205
MSC: Primary 20H15, 55M20; Secondary 20C10, 20F34.
DOI: 10.4064/fm223-3-1
Abstract
Let $M$ be a flat manifold. We say that $M$ has the $R_\infty $ property if the Reidemeister number $R(f)$ is infinite for every homeomorphism $f : M \to M.$ We investigate relations between the holonomy representation $\rho $ of $M$ and the $R_\infty $ property. 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 the $R_\infty $ property. This result is related to Conjecture 4.8 in [K. Dekimpe et al., Topol. Methods Nonlinear Anal. 34 (2009)].
