Twisted conjugacy classes in nilpotent groups

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Gonçalves, Daciberg
Wong, Peter
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Abstract
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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.
Comment: 22 pages; section 6 has been moved to section 2 and minor modification has been made on exposition; to be published in Crelle J
Keywords
Mathematics - Group Theory, Mathematics - Algebraic Topology, 20E45, 55M20
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