TWISTED CONJUGACY CLASSES IN CHEVALLEY GROUPS

Let G be a group and phi : G -> G its automorphism. We say that elements x and y of G are twisted phi conjugate, or merely phi conjugate (written x similar to(phi) y), if there exists an element z of G for which x = zy phi(z(-1)). If, in addition, phi is an identical automorphism, then we speak of conjugacy. The phi conjugacy class of an element x is denoted by [x](phi). The number R(phi) of these classes is called the Reidemeister number of an automorphism phi. A group is said to possess the R-infinity property if the number R(.) is infinite for every automorphism phi. We consider Chevalley groups over fields. In particular, it is proved that if an algebraically closed field F of characteristic zero has finite transcendence degree over Q, then a Chevalley group over F possesses the R-infinity property. Furthermore, a Chevalley group over a field F of characteristic zero has the R-infinity property if F has a periodic automorphism group. The condition that F is of characteristic zero cannot be discarded. This follows from Steinberg's result which says that for connected linear algebraic groups over an algebraically closed field of characteristic zero, there always exists an automorphism phi for which R(phi) = 1.