Twisted conjugacy in ${\mathrm{SL}}_n$ and ${\mathrm{GL}}_n$ over subrings of $\overline{{\mathbb{F}}_p}(t)$

Abstract

Let $\phi :G\to G$ be an automorphism of an infinite group $G$. One has an equivalence relation ${\sim }_{\phi }$ on $G$ defined as $x{\sim }_{\phi }y$ if there exists a $z\in G$ such that $y=zx\phi (z^{-1})$. The equivalence classes are called $\phi$-twisted conjugacy classes, and the set $G/{\sim }_{\phi }$ of equivalence classes is denoted by $\mathcal{R}(\phi )$. The cardinality $R(\phi )$ of $\mathcal{R}(\phi )$ is called the Reidemeister number of $\phi$. We write $R(\phi )=\infty$ when $\mathcal{R}(\phi )$ is infinite. We say that $G$ has the $R_{\infty }$-property if $R(\phi )=\infty$ for every automorphism $\phi$ of $G$. We show that the groups $G={\mathrm{GL}}_n(R),{\mathrm{SL}}_n(R)$ have the $R_{\infty }$-property for all $n\ge 3$ when $F[t]\subset R⊊F(t)$, where $F$ is a subfield of $\overline{{\mathbb{F}}_p}$. When $n\ge 4$, we show that any subgroup $H\subset {\mathrm{GL}}_n(R)$ that contains ${\mathrm{SL}}_n(R)$ also has the $R_{\infty }$-property.