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Automorphisms of the Gersten Group

Authors
  • Dudkin, F. A.1, 2
  • Shaporina, E. A.3
  • 1 Sobolev Institute of Mathematics, Novosibirsk, Russia , Novosibirsk (Russia)
  • 2 Omsk Division of the Sobolev Institute of Mathematics, Omsk, Russia , Omsk (Russia)
  • 3 Novosibirsk State University, Novosibirsk, Russia , Novosibirsk (Russia)
Type
Published Article
Journal
Siberian Mathematical Journal
Publisher
Pleiades Publishing
Publication Date
May 27, 2021
Volume
62
Issue
3
Pages
413–422
Identifiers
DOI: 10.1134/S0037446621030046
Source
Springer Nature
Keywords
License
Yellow

Abstract

The Gersten group \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ G $\end{document} is the split extension \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ F_{3}\rtimes_{\varphi}{𝕑} $\end{document} of the free group \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ F_{3} $\end{document} with basis \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \{a,b,c\} $\end{document} by the automorphism \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \varphi:a\mapsto a,b\mapsto ba,c\mapsto ca^{2} $\end{document}. We describe the generators and structure of the group \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \operatorname{Out}(G) $\end{document} and prove that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \operatorname{Out}(G)\cong(F_{3}\times{𝕑}^{3})\rtimes({𝕑}_{2}\times{𝕑}_{2}) $\end{document}.

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