Twisted conjugacy classes in nilpotent groups

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 $R_\infty$ 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.