Orbit decidability and the conjugacy problem for some extensions of groups
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- by O. Bogopolski, A. Martino and E. Ventura PDF
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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.References
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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
- MR Author ID: 646503
- Email: A.Martino@soton.ac.uk
- E. Ventura
- Affiliation: Departament de Matemàtica Aplicada III, Universitat Politècnica de Catalunya, Barcelona, Catalonia, Spain
- MR Author ID: 340143
- Email: enric.ventura@upc.edu
- 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: https://doi.org/10.1090/S0002-9947-09-04817-X
- MathSciNet review: 2574885