On the Reidemeister spectrum and the $R_{\infty}$ property for some free nilpotent groups

TL;DR

This paper analyzes the Reidemeister spectrum and the $R_{ olinebreak _{ olinebreak ext{infty}}}$ property for free nilpotent and solvable groups, identifying specific cases with the $R_{ olinebreak _{ olinebreak ext{infty}}}$ property and describing the spectrum.

Contribution

It provides a detailed description of the Reidemeister spectrum for certain free nilpotent groups and establishes the $R_{ olinebreak _{ olinebreak ext{infty}}}$ property for large classes of free solvable groups.

Findings

Reidemeister spectrum explicitly described for some free nilpotent groups.

Groups $N_{2c}$ with $c \\geq 4$ satisfy the $R_{ olinebreak _{ olinebreak ext{infty}}}$ property.

All free solvable groups $S_{2t}$ of rank 2 and class $t \\geq 2$ satisfy the $R_{ olinebreak _{ olinebreak ext{infty}}}$ property.

Abstract

We describe the Reidemeister spectrum $Spec_RG$ for $G = N_{rc},$ the free nilpotent group of rank $r$ and class $c,$ in the cases: $r \in {\mathbb N}$ and $c = 1;$ $r = 2, 3$ and $c = 2;$ $ r = 2$ and $c = 3,$ and prove that any group $N_{2c}$ for $c \geq 4$ satisfies to the $R_{\infty}$ property. As a consequence we obtain that every free solvable group $S_{2t}$ of rank 2 and class $t \geq 2$ (in particular the free metabelian group $M_2 = S_{22}$ of rank 2) satisfies to the $R_{\infty}$ property. Moreover, we prove that any free solvable group $S_{rt}$ of rank $r \geq 2$ and class $t$ big enough also satisfies to the $R_{\infty}$ property.