Twisted conjugacy classes in twisted Chevalley groups

Abstract

Let G be a group and φ be an automorphism of G. Two elements x,y∈G are said to be φ-twisted conjugate if y=gxφ(g)−1 for some g∈G. We say that a group G has the R∞-property if the number of φ-twisted conjugacy classes is infinite for every automorphism φ of G. In this paper, we prove that twisted Chevalley groups over a field k of characteristic zero have the R∞-property as well as the S∞-property if k has finite transcendence degree over Q or Aut(k) is periodic.