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

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.