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 → M. We investigate relations between the holonomy representation ρ of M and the $R_{\infty }$ property. When the holonomy group of M is solvable we show that if ρ has a unique 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)].

We give a criterion which ensures that a group generated by Cartan involutions in the automorph group of a rational quadratic form of signature $(n-1,1)$ is “thin”, namely it is of infinite index in the latter. It is based on a graph defined on the integral Cartan root vectors, as well as Vinberg’s theory of hyperbolic reflection groups. The criterion is shown to be robust for showing that many hyperbolic hypergeometric groups for ${}_n{F}_{n-1}$ are thin.