Twisted conjugacy in $SL_n$ and $GL_n$ over subrings of $\bar{\mathbb F}_p(t)$

TL;DR

This paper proves that certain linear groups over subrings of algebraic function fields have the property that all their automorphisms have infinitely many twisted conjugacy classes, extending understanding of their algebraic and dynamical properties.

Contribution

It establishes the $R_ty$-property for $GL_n(R)$ and $SL_n(R)$ over specific subrings of $ar{F}_p(t)$ for all $n eq 2$, and for subgroups containing $SL_n(R)$ when $n eq 3$.

Findings

Groups $GL_n(R)$ and $SL_n(R)$ have the $R_ty$-property for $n eq 2$.

Subgroups containing $SL_n(R)$ also have the $R_ty$-property for $n eq 3$.

The results apply to rings between polynomial and rational functions over subfields of algebraic closures.

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 $\mathcal R(\phi)$. 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 $R_\infty$-{\it property} if $R(\phi)=\infty$ for every automorphism $\phi$ of $G$. We show that the groups $G=GL_n(R), SL_n(R)$ have the $R_\infty$-property for all $n\ge 3$ when $ F[t]\subset R\subsetneq F(t)$ where $F$ is a subfield of $\bar{\mathbb F}_p$. When $n\ge 4$, we show that any subgroup $H\subset GL_n(R)$ that contains $SL_n(R)$ also has the $R_\infty$-property.