The University of Southampton
University of Southampton Institutional Repository

Orbit decidability and the conjugacy problem for some extensions of groups

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.
0002-9947
2003-2036
Bogopolski, Oleg
66c17a75-5fe6-48b3-a77b-ea071c4c00db
Martino, Armando
65f1ff81-7659-4543-8ee2-0a109be286f1
Ventura, Enric
543ad8f8-4af2-41c5-91ec-7781d79bf647
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), 2003-2036.

Record type: Article

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.

PDF
0712.3104v1.pdf - Accepted Manuscript
Download (421kB)
PDF
__userfiles.soton.ac.uk_Users_nsc_mydesktop_69565martino.pdf - Version of Record
Restricted to Repository staff only
Request a copy

More information

Published date: 2009
Organisations: Pure Mathematics

Identifiers

Local EPrints ID: 69565
URI: https://eprints.soton.ac.uk/id/eprint/69565
ISSN: 0002-9947
PURE UUID: a621e58e-4cf5-40d5-9f39-6b4a0548d51e
ORCID for Armando Martino: ORCID iD orcid.org/0000-0002-5350-3029

Catalogue record

Date deposited: 19 Nov 2009
Last modified: 13 Jun 2019 00:33

Export record

Download statistics

Downloads from ePrints over the past year. Other digital versions may also be available to download e.g. from the publisher's website.

View more statistics

Atom RSS 1.0 RSS 2.0

Contact ePrints Soton: eprints@soton.ac.uk

ePrints Soton supports OAI 2.0 with a base URL of https://eprints.soton.ac.uk/cgi/oai2

This repository has been built using EPrints software, developed at the University of Southampton, but available to everyone to use.

We use cookies to ensure that we give you the best experience on our website. If you continue without changing your settings, we will assume that you are happy to receive cookies on the University of Southampton website.

×