Twisted conjugacy classes in Chevalley groups

TL;DR

This paper proves that Chevalley groups over certain fields have the $R_{ ext{infinity}}$ property, and characterizes when their twisted conjugacy class of the identity element forms a subgroup.

Contribution

It establishes the $R_{ ext{infinity}}$ property for Chevalley groups over specific fields and characterizes central automorphisms via twisted conjugacy classes.

Findings

Chevalley groups over fields with torsion automorphisms have $R_{ ext{infinity}}$ property.

The twisted conjugacy class of the identity is a subgroup iff the automorphism is central.

The results apply to algebraically closed fields with finite transcendence degree over $Q$.

Abstract

We prove that Chevalley group over the field $F$ of zero characteristic possess $R_{\infty}$ property, if $F$ has torsion group of automorphisms or $F$ is an algebraically closed field which has finite transcendence degree over $\mathbb{Q}$. As a consequence we obtain that the twisted conjugacy class $[e]_{\varphi}$ of unit element is a subgroup of Chevalley group if and only if $\varphi$ is central automorphism.