Twisted Conjugacy in Linear Algebraic Groups II

TL;DR

This paper investigates the algebraic $R_ abla$-property in linear algebraic groups, establishing conditions under which the property holds or fails, with implications for automorphism fixed points and subgroup structures.

Contribution

It proves that the algebraic $R_ abla$-property is equivalent to the infinitude of fixed points under automorphisms, and shows Borel subgroups of semisimple groups possess this property.

Findings

Connected non-solvable groups have the $R_ abla$-property.

Borel subgroups of semisimple groups have the $R_ abla$-property.

Certain solvable groups do not have the $R_ abla$-property.

Abstract

Let $G$ be a linear algebraic group over an algebraically closed field $k$ and $\mathrm{Aut}_{\mathrm{alg}}(G)$ the group of all algebraic group automorphisms of $G$. For every $\varphi\in \mathrm{Aut}_{\mathrm{alg}}(G)$ let $\mathcal{R}(\varphi)$ denote the set of all orbits of the $\varphi$-twisted conjugacy action of $G$ on itself (given by $(g,x)\mapsto gx\varphi(g^{-1})$, for all $g,x\in G$). We say that $G$ has the algebraic $R_\infty$-property if $\mathcal{R}(\varphi)$ is infinite for every $\varphi\in \mathrm{Aut}_{\mathrm{alg}}(G)$. In \citep{bb} we have shown that this property is satisfied by every connected non-solvable algebraic group. From a theorem due to Steinberg it follows that if a connected algebraic group $G$ has the algebraic $R_\infty$-property, then $G^\varphi$ (the fixed-point subgroup of $G$ under $\varphi$) is infinite for all $\varphi\in \mathrm{Aut}_{\mathrm{alg}}(G)$. In this article we show that the condition is also sufficient. We also show that a Borel subgroup of any semisimple algebraic group has the algebraic $R_\infty$-property and identify certain classes of solvable algebraic groups for which the property fails.