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Transactions of the American Mathematical Society

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Orbit decidability and the conjugacy problem for some extensions of groups


Authors: O. Bogopolski, A. Martino and E. Ventura
Journal: Trans. Amer. Math. Soc. 362 (2010), 2003-2036
MSC (2000): Primary 20F10
DOI: https://doi.org/10.1090/S0002-9947-09-04817-X
Published electronically: November 16, 2009
MathSciNet review: 2574885
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Abstract: 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: Oleg_Bogopolski@yahoo.com

A. Martino
Affiliation: School of Mathematics, University of Southampton, Southampton, England
Email: A.Martino@soton.ac.uk

E. Ventura
Affiliation: Departament de Matemàtica Aplicada III, Universitat Politècnica de Catalunya, Barcelona, Catalonia, Spain
Email: enric.ventura@upc.edu

DOI: https://doi.org/10.1090/S0002-9947-09-04817-X
Received by editor(s): December 19, 2007
Published electronically: November 16, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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