Twisted conjugacy classes in nilpotent groups

A group is said to have the $R_{\mathrm{\infty }}$ property if every automorphism has an infinite number of twisted conjugacy classes. We study the question whether $G$ has the $R_{\mathrm{\infty }}$ property when $G$ is a finitely generated torsion-free nilpotent group. As a consequence, we show that for every positive integer $n\ge 5$, there is a compact nilmanifold of dimension $n$ on which every homeomorphism is isotopic to a fixed point free homeomorphism. As a by-product, we give a purely group theoretic proof that the free group on two generators has the $R_{\mathrm{\infty }}$ property. The $R_{\mathrm{\infty }}$ property for virtually abelian and for $\mathcal{C}$-nilpotent groups are also discussed.