Twisted conjugacy classes in nilpotent groups
Daciberg Gon\c{c}alves, Peter Wong

TL;DR
This paper investigates the $R_ty$ property in finitely generated torsion-free nilpotent groups, showing its implications for nilmanifolds and providing new proofs for known cases, expanding understanding of twisted conjugacy classes.
Contribution
It establishes the $R_ty$ property for certain nilpotent groups, connects this to topological properties of nilmanifolds, and offers a new group-theoretic proof for free groups.
Findings
Finitely generated torsion-free nilpotent groups can have the $R_ty$ property.
Every homeomorphism on certain high-dimensional nilmanifolds is isotopic to a fixed point free homeomorphism.
The free group on two generators has the $R_ty$ property, proven via group theory.
Abstract
A group is said to have the property if every automorphism has an infinite number of twisted conjugacy classes. We study the question whether has the property when is a finitely generated torsion-free nilpotent group. As a consequence, we show that for every positive integer , there is a compact nilmanifold of dimension 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 property. The property for virtually abelian and for -nilpotent groups are also discussed.
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