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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Orbit decidability and the conjugacy problem for some extensions of groups
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by O. Bogopolski, A. Martino and E. Ventura PDF
Trans. Amer. Math. Soc. 362 (2010), 2003-2036 Request permission


Given a short exact sequence of groups with certain conditions, $1\rightarrow F\rightarrow G\rightarrow H\rightarrow 1$, we prove that $G$ has solvable conjugacy problem if and only if the corresponding action subgroup $A\leqslant Aut(F)$ is orbit decidable. From this, we deduce that the conjugacy problem is solvable, among others, for all groups of the form $\mathbb {Z}^2\rtimes F_m$, $F_2\rtimes F_m$, $F_n \rtimes \mathbb {Z}$, and $\mathbb {Z}^n \rtimes _A F_m$ with virtually solvable action group $A\leqslant GL_n(\mathbb {Z})$. Also, we give an easy way of constructing groups of the form $\mathbb {Z}^4\rtimes F_n$ and $F_3\rtimes F_n$ with unsolvable conjugacy problem. On the way, we solve the twisted conjugacy problem for virtually surface and virtually polycyclic groups, and we give an example of a group with solvable conjugacy problem but unsolvable twisted conjugacy problem. As an application, an alternative solution to the conjugacy problem in $Aut(F_2)$ is given.
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Additional Information
  • O. Bogopolski
  • Affiliation: Institute of Mathematics, Siberian Branch of The Russian Academy of Sciences, Novosibirsk, Russia
  • Address at time of publication: Mathematisches Institut der Heinrich-Heine-Universität Düsseldorf, Düsseldorf, Germany
  • Email:
  • A. Martino
  • Affiliation: School of Mathematics, University of Southampton, Southampton, England
  • MR Author ID: 646503
  • Email:
  • E. Ventura
  • Affiliation: Departament de Matemàtica Aplicada III, Universitat Politècnica de Catalunya, Barcelona, Catalonia, Spain
  • MR Author ID: 340143
  • Email:
  • Received by editor(s): December 19, 2007
  • Published electronically: November 16, 2009
  • © Copyright 2009 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Trans. Amer. Math. Soc. 362 (2010), 2003-2036
  • MSC (2000): Primary 20F10
  • DOI:
  • MathSciNet review: 2574885