Abstract

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