Reidemeister spectrum of special and general linear groups over some fields contains 1

TL;DR

This paper investigates the Reidemeister spectrum of special and general linear groups over certain fields, showing that for fields with infinite transcendence degree, these groups can have automorphisms with Reidemeister number 1, unlike the finite transcendence degree case.

Contribution

It demonstrates the existence of automorphisms with Reidemeister number 1 for ${ m SL}_n( extbf{F})$ and ${ m GL}_n( extbf{F})$ over fields with infinite transcendence degree, contrasting with the finite case.

Findings

Existence of automorphisms with Reidemeister number 1 over fields with infinite transcendence degree.

${ m SL}_n( extbf{F})$ and ${ m GL}_n( extbf{F})$ do not have the $R_{ extinfty}$-property in this setting.

Finite transcendence degree fields lead to these groups having the $R_{ extinfty}$-property.

Abstract

We prove that if $\mathbb{F}$ is an algebraically closed field of zero characteristic which has infinite transcendence degree over $\mathbb{Q}$, then there exists a field automorphism $\varphi$ of ${\rm SL}_n(\mathbb{F})$ and ${\rm GL}_n(\mathbb{F})$ such that $R(\varphi)=1$. This fact implies that ${\rm SL}_n(\mathbb{F})$ and ${\rm GL}_n(\mathbb{F})$ do not possess the $R_{\infty}$-property. However, if the transcendece degree of $\mathbb{F}$ over $\mathbb{Q}$ is finite, then ${\rm SL}_n(\mathbb{F})$ and ${\rm GL}_n(\mathbb{F})$ are known to possess the $R_{\infty}$-property.