Twisted Conjugacy in Linear Algebraic Groups II

DOI10.1016/J.JALGEBRA.2022.03.031arXiv2106.04242MaRDI QIDQ6369744

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Publication date: 8 June 2021

Abstract: Let $G$ be a linear algebraic group over an algebraically closed field $k$ and $mathrm{Aut}_{mathrmalg}(G)$ the group of all algebraic group automorphisms of $G$. For every $varphiinmathrm{Aut}_{mathrmalg}(G)$ let $mathcalR(varphi)$ denote the set of all orbits of the $varphi$-twisted conjugacy action of $G$ on itself (given by $(g,x)mapstogxvarphi(g^{-1})$, for all $g,xinG$). We say that $G$ has the algebraic $R_{i}nfty$-property if $mathcalR(varphi)$ is infinite for every $varphiinmathrm{Aut}_{mathrmalg}(G)$. 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 $R_{i}nfty$-property, then $G^{v}arphi$ (the fixed-point subgroup of $G$ under $varphi$) is infinite for all $varphiinmathrm{Aut}_{mathrmalg}(G)$. 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 $R_{i}nfty$-property and identify certain classes of solvable algebraic groups for which the property fails.

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