Twisted conjugacy classes in nilpotent groups

Abstract

A group is said to have the $R_{\infty }$ property if every automorphism has an infinite number of twisted conjugacy classes. We study the question whether $G$ has the $R_{\infty }$ 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 $R_{\infty }$ property. The $R_{\infty }$ property for virtually abelian and for $\mathcal{C}$-nilpotent groups are also discussed.

arXiv comment22 pages; section 6 has been moved to section 2 and minor modification has been made on exposition; to be published in Crelle J