Twisted conjugacy in linear algebraic groups

Abstract

Let $k$ be an algebraically closed field, $G$ a linear algebraic group over $k$ and $\phi \in Aut(G)$, the group of all algebraic group automorphisms of $G$. Two elements $x,y$ of $G$ are said to be $\phi$-twisted conjugate if $y=gx\phi (g{)}^{-1}$ for some $g\in G$. In this paper we prove that for a connected non-solvable linear algebraic group $G$ over $k$, the number of its $\phi$-twisted conjugacy classes is infinite for every $\phi \in Aut(G)$.

15 pages, 1 figure. Some proofs are rewritten. Final version to appear in Transformation Groups