Twisted Conjugacy in Linear Algebraic Groups

Abstract

Let G be a linear algebraic group over an algebraically closed field k and Aut(G) the group of all algebraic group automorphisms of G. For every ϕ ∈ Aut(G) let R(ϕ) denote the set of all orbits of the ϕ-twisted conjugacy action of G on itself (given by (g, x) → gxϕ(g −1 ), for all g, x ∈ G). We say that G satisfies the R ∞ -property if R(ϕ) is infinite for every ϕ ∈ Aut(G). In [BB21], 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 R ∞ -property then G ϕ is infinite for all ϕ ∈ Aut(G). In this article we show that the condition is also sufficient. In particular we deduce that a Borel subgroup of any semisimple algebraic group has the R ∞ -property and identify certain classes of solvable algebraic groups for which the property fails.