The University of Southampton
University of Southampton Institutional Repository

# Orbit decidability and the conjugacy problem for some extensions of groups

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.

Text
0712.3104v1.pdf - Accepted Manuscript
Text
__userfiles.soton.ac.uk_Users_nsc_mydesktop_69565martino.pdf - Version of Record
Restricted to Repository staff only
Request a copy

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
ORCID for Armando Martino: orcid.org/0000-0002-5350-3029

## Catalogue record

Date deposited: 19 Nov 2009

## Contributors

Author: Oleg Bogopolski
Author: Armando Martino
Author: Enric Ventura