Twisted conjugacy classes in nilpotent groups

A group is said to have the R-infinity property if every automorphism has an infinite number of twisted conjugacy classes. We study the question whether G has the R-infinity property when G is a finitely generated torsion-free nilpotent group. As a consequence, we show that for every positive integer n >= 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-infinity property. The R-infinity property for virtually abelian and for C-nilpotent groups are also discussed.