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 Algebra Logika, 2014, Volume 53, Number 6, Pages 735–763 (Mi al664)

Twisted conjugacy classes in Chevalley groups

T. R. Nasybullov

Novosibirsk State University, ul. Pirogova 2, Novosibirsk, 630090, Russia

Abstract: Let $G$ be a group and $\varphi\colon G\to G$ its automorphism. We say that elements $x$ and $y$ of $G$ are twisted $\varphi$-conjugate, or merely $\varphi$-conjugate (written $x\sim_\varphi y$), if there exists an element $z$ of $G$ for which $x=zy\varphi(z^{-1})$. If, in addition, $\varphi$ is an identical automorphism, then we speak of conjugacy. The $\varphi$-conjugacy class of an element $x$ is denoted by $[x]_\varphi$. The number $R(\varphi)$ of these classes is called the Reidemeister number of an automorphism $\varphi$. A group is said to possess the $R_\infty$ property if the number $R(\varphi)$ is infinite for every automorphism $\varphi$.
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 $\mathbb Q$, then a Chevalley group over $F$ possesses the $R_\infty$ property. Furthermore, a Chevalley group over a field $F$ of characteristic zero has the $R_\infty$ 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 $\varphi$ for which $R(\varphi)=1$.

Keywords: twisted conjugacy classes, Chevalley group.

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English version:
Algebra and Logic, 2015, 53:6, 481–501

Bibliographic databases:

UDC: 512.54
Revised: 24.07.2014

Citation: T. R. Nasybullov, “Twisted conjugacy classes in Chevalley groups”, Algebra Logika, 53:6 (2014), 735–763; Algebra and Logic, 53:6 (2015), 481–501

Citation in format AMSBIB
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\by T.~R.~Nasybullov
\paper Twisted conjugacy classes in Chevalley groups
\jour Algebra Logika
\yr 2014
\vol 53
\issue 6
\pages 735--763
\mathnet{http://mi.mathnet.ru/al664}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3408307}
\transl
\jour Algebra and Logic
\yr 2015
\vol 53
\issue 6
\pages 481--501
\crossref{https://doi.org/10.1007/s10469-015-9310-4}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84924166817}

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. Fel'shtyn, E. Troitsky, “Aspects of the property $R_{\infty}$”, J. Group Theory, 18:6 (2015), 1021–1034
2. T. R. Nasybullov, “The $R_{\infty}$-property for chevalley groups of types $B_l$, $C_l$, $D_l$ over integral domains”, J. Algebra, 446 (2016), 489–498
3. A. Fel'shtyn, T. Nasybullov, “The $R_{\infty}$ and $S_{\infty}$ properties for linear algebraic groups”, J. Group Theory, 19:5 (2016), 901–921
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