Twisted conjugacy classes in twisted Chevalley groups
Sushil Bhunia, Pinka Dey, Amit Roy
TL;DR
The paper addresses when twisted Chevalley groups $G'_{\sigma}$ over characteristic-zero fields possess the $R_{\infty}$-property and the $S_{\infty}$-property. It extends the Reidemeister-number framework to twisted Chevalley groups by leveraging Steinberg's decomposition of automorphisms into inner, diagonal, and field parts, and constructs explicit families of elements to force infinitely many twisted conjugacy classes under any automorphism when the base field has finite transcendence degree or periodic automorphism group. A key contribution is the characterization of when the $\varphi$-twisted conjugacy class of the identity, $[e]_{\varphi}$, is a subgroup, showing this occurs precisely for central automorphisms, thereby linking $R_{\infty}$ and $S_{\infty}$ properties in these groups. The results broaden the understanding of twisted conjugacy phenomena in algebraic groups, providing groundwork for further extensions to rings and broader field-parameter regimes, and connect to prior work by Nasybullov and Fel'shtyn on untwisted Chevalley groups.
Abstract
Let G be a group and φ be an automorphism of G. Two elements x, y of G are said to be φ-twisted if y = gxφ(g)^{-1} for some g in G. We say that a group G has the R_{\infty}-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 the field k of characteristic zero have the R_{\infty}-property as well as S_{\infty}-property if k has finite transcendence degree over \mathbb{Q} or Aut(k) is periodic.