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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## More information

Published date: 2009

Organisations:
Pure Mathematics

## Identifiers

Local EPrints ID: 69565

URI: http://eprints.soton.ac.uk/id/eprint/69565

ISSN: 0002-9947

PURE UUID: a621e58e-4cf5-40d5-9f39-6b4a0548d51e

## Catalogue record

Date deposited: 19 Nov 2009

Last modified: 14 Mar 2020 01:31

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## Contributors

Author:
Oleg Bogopolski

Author:
Enric Ventura

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