Orbit decidability and the conjugacy problem for some extensions of groups
Orbit decidability and the conjugacy problem for some extensions of groups
Given a short exact sequence of groups with certain conditions, 1 ? F ? G ? H ? 1, we
prove that G has solvable conjugacy problem if and only if the corresponding action subgroup
A 6 Aut(F) is orbit decidable. From this, we deduce that the conjugacy problem is solvable,
among others, for all groups of the form Z2?Fm, F2?Fm, Fn?Z, and Zn?A Fm with virtually solvable action group A 6 GLn(Z). Also, we give an easy way of constructing groups of the form Z4?Fn and F3?Fn with unsolvable conjugacy problem. On the way, we solve the twisted conjugacy problem for virtually surface and virtually polycyclic groups, and 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(F2) is given.
2003-2036
Bogopolski, Oleg
66c17a75-5fe6-48b3-a77b-ea071c4c00db
Martino, Armando
65f1ff81-7659-4543-8ee2-0a109be286f1
Ventura, Enric
543ad8f8-4af2-41c5-91ec-7781d79bf647
2009
Bogopolski, Oleg
66c17a75-5fe6-48b3-a77b-ea071c4c00db
Martino, Armando
65f1ff81-7659-4543-8ee2-0a109be286f1
Ventura, Enric
543ad8f8-4af2-41c5-91ec-7781d79bf647
Bogopolski, Oleg, Martino, Armando and Ventura, Enric
(2009)
Orbit decidability and the conjugacy problem for some extensions of groups.
Transactions of the American Mathematical Society, 362 (4), .
Abstract
Given a short exact sequence of groups with certain conditions, 1 ? F ? G ? H ? 1, we
prove that G has solvable conjugacy problem if and only if the corresponding action subgroup
A 6 Aut(F) is orbit decidable. From this, we deduce that the conjugacy problem is solvable,
among others, for all groups of the form Z2?Fm, F2?Fm, Fn?Z, and Zn?A Fm with virtually solvable action group A 6 GLn(Z). Also, we give an easy way of constructing groups of the form Z4?Fn and F3?Fn with unsolvable conjugacy problem. On the way, we solve the twisted conjugacy problem for virtually surface and virtually polycyclic groups, and 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(F2) is given.
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Published date: 2009
Organisations:
Pure Mathematics
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Local EPrints ID: 69565
URI: http://eprints.soton.ac.uk/id/eprint/69565
ISSN: 0002-9947
PURE UUID: a621e58e-4cf5-40d5-9f39-6b4a0548d51e
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Date deposited: 19 Nov 2009
Last modified: 14 Mar 2024 02:54
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Author:
Oleg Bogopolski
Author:
Enric Ventura
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