Twisted conjugacy in linear algebraic groups II

Abstract

Let G be a linear algebraic group over an algebraically closed field k and the group of all algebraic group automorphisms of G. For every let denote the set of all orbits of the φ-twisted conjugacy action of G on itself (given by , for all ). We say that G has the algebraic -property if is infinite for every . In [1] 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 -property, then (the fixed-point subgroup of G under φ) is infinite for all . 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 -property and identify certain classes of solvable algebraic groups for which the property fails.